HSPT Math Quiz: Apply Pythagorean Theorem

What this quiz covers

This quiz focuses on Apply Pythagorean Theorem, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $6\sqrt{2}\text{ cm}$
2. $6\sqrt{3}\text{ cm}$
3. $9\text{ cm}$
4. $12\text{ cm}$ (correct answer)

Explanation: When you encounter a $30°\text{–}60°\text{–}90°$ triangle, you're dealing with one of geometry's most predictable special right triangles. These triangles have fixed side ratios that make calculations straightforward once you memorize the pattern. In any $30°\text{–}60°\text{–}90°$ triangle, the sides are always in the ratio $1 : \sqrt{3} : 2$. Specifically:

• The shortest side (opposite the $30°$ angle) has length $x$
• The longer leg (opposite the $60°$ angle) has length $x\sqrt{3}$
• The hypotenuse (opposite the $90°$ angle) has length $2x$

Question 2

1. 15 cm
2. $\sqrt{145}$ cm
3. 17 cm (correct answer)
4. 29 cm

Explanation: The length of the space diagonal (d) of a rectangular prism with length l, width w, and height h is given by the formula $d^2 = l^2 + w^2 + h^2$, which is an application of the Pythagorean theorem in three dimensions. Substitute the given values: $d^2 = 12^2 + 9^2 + 8^2$. $d^2 = 144 + 81 + 64$. $d^2 = 289$. Taking the square root of both sides, $d = \sqrt{289} = 17$ cm.

Question 3

1. 15 meters
2. 20 meters (correct answer)
3. 30 meters
4. $5\sqrt{34}$ meters

Explanation: Imagine a right triangle formed by the two poles and the cable. The horizontal leg is the distance between the poles (which we need to find). The vertical leg is the difference in the heights of the poles. The cable is the hypotenuse. The difference in heights is $35 - 20 = 15$ meters. The length of the hypotenuse is 25 meters. Let the distance between the bases be $d$. By the Pythagorean theorem: $d^2 + 15^2 = 25^2$. $d^2 + 225 = 625$. $d^2 = 625 - 225 = 400$. $d = \sqrt{400} = 20$ meters. This is a multiple of a 3-4-5 right triangle (15-20-25).

Question 4

1. 15
2. 7.5
3. 7.2 (correct answer)
4. 6.8

Explanation: First, find the length of the hypotenuse FH using the Pythagorean theorem: $FG^2 + GH^2 = FH^2$. $9^2 + 12^2 = FH^2$. $81 + 144 = FH^2$. $225 = FH^2$, so $FH = 15$. (This is a 3-4-5 triangle scaled by 3). The area of the triangle can be calculated in two ways: using the legs as base and height, or using the hypotenuse as the base and the altitude GK as the height. Area = $\frac{1}{2} \times FG \times GH = \frac{1}{2} \times 9 \times 12 = 54$. Also, Area = $\frac{1}{2} \times FH \times GK = \frac{1}{2} \times 15 \times GK$. Setting the two expressions for the area equal: $54 = \frac{1}{2} \times 15 \times GK$. $108 = 15 \times GK$. $GK = \frac{108}{15} = \frac{36}{5} = 7.2$.

Question 5

1. 4 feet
2. 8 feet (correct answer)
3. 15 feet
4. 11 feet

Explanation: This is a two-step problem. First, find the initial height of the ladder on the wall. Let the height be $h_1$. The ladder, wall, and ground form a right triangle. $h_1^2 + 7^2 = 25^2$. $h_1^2 + 49 = 625$. $h_1^2 = 576$, so $h_1 = 24$ feet. The top slides down 4 feet, so the new height is $h_2 = 24 - 4 = 20$ feet. Now, find the new distance of the base from the wall, $b_2$. The ladder's length remains 25 feet. $20^2 + b_2^2 = 25^2$. $400 + b_2^2 = 625$. $b_2^2 = 225$, so $b_2 = 15$ feet. The question asks how much farther the base slides, which is the difference between the new and old distances: $15 - 7 = 8$ feet.

Question 6

1. 10 cm
2. 11 cm
3. 12 cm (correct answer)
4. 14 cm

Explanation: When you encounter a right circular cone problem involving radius, slant height, and height, you're working with the Pythagorean theorem. The cone's height, base radius, and slant height form a right triangle where the height is one leg, the radius is the other leg, and the slant height is the hypotenuse. Given a base radius of 5 cm and slant height of 13 cm, you can find the height using: $h^2 + r^2 = s^2$, where h is height, r is radius, and s is slant height. Substituting the values: $h^2 + 5^2 = 13^2$, so $h^2 + 25 = 169$. Therefore, $h^2 = 144$, and $h = 12$ cm. Let's examine why the other answers are incorrect: A) 10 cm would give us $10^2 + 5^2 = 100 + 25 = 125$, but $13^2 = 169$. This doesn't satisfy the Pythagorean theorem. B) 11 cm would yield $11^2 + 5^2 = 121 + 25 = 146$, which is still less than 169. D) 14 cm would produce $14^2 + 5^2 = 196 + 25 = 221$, which exceeds 169. Only C) 12 cm correctly satisfies the relationship: $12^2 + 5^2 = 144 + 25 = 169 = 13^2$. Study tip: Remember that in cone problems, you're often dealing with a right triangle formed by the height, radius, and slant height. Always check which measurement you're missing and apply the Pythagorean theorem accordingly. The 5-12-13 triangle is a common Pythagorean triple worth memorizing.

Question 7

1. 32
2. 40 (correct answer)
3. 48
4. 56

Explanation: When you encounter a rhombus with given diagonal lengths, remember that the diagonals of a rhombus are perpendicular and bisect each other. This creates four congruent right triangles within the rhombus, which is the key to finding the side length. Since the diagonals have lengths 16 and 12, each diagonal is split in half at their intersection point. This gives you right triangles with legs of length 8 (half of 16) and 6 (half of 12). The hypotenuse of each right triangle is a side of the rhombus. Using the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a = 8$ and $b = 6$: $8^2 + 6^2 = c^2$ $64 + 36 = c^2$ $100 = c^2$ $c = 10$ Since all four sides of a rhombus are equal, the perimeter is $4 \times 10 = 40$. Looking at the wrong answers: Choice A (32) likely comes from adding the diagonal lengths (16 + 12 = 28) and rounding or making an arithmetic error. Choice C (48) might result from incorrectly using the full diagonal lengths as legs in the Pythagorean theorem, giving $\sqrt{16^2 + 12^2} = 20$, then doubling instead of quadrupling. Choice D (56) could come from adding the diagonals and then adding the calculated side length (28 + 10 + 10 + 8). Remember: when given diagonal lengths of a rhombus, always halve them first to find the legs of the right triangles formed at the intersection point.

Question 8

1. $\sqrt{1225} + 20$ meters
2. $25\sqrt{2} + 20$ meters
3. $\sqrt{850} + 20$ meters
4. $5\sqrt{34} + 20$ meters (correct answer)

Explanation: When you encounter a problem involving diagonal distances in rectangles, you're working with the Pythagorean theorem. This question has two parts: finding the diagonal across the pool, then adding the extra distance. To find the diagonal of the rectangular pool, you need to use $a^2 + b^2 = c^2$ where the length and width are the legs, and the diagonal is the hypotenuse. With a pool that's 25 meters long and 15 meters wide: $25^2 + 15^2 = c^2$ $625 + 225 = 850$ $c = \sqrt{850}$ You can simplify $\sqrt{850}$ by factoring: $850 = 25 \times 34 = 5^2 \times 34$, so $\sqrt{850} = 5\sqrt{34}$. The rope then continues 20 meters beyond the pool, so the total length is $5\sqrt{34} + 20$ meters. Looking at the wrong answers: Choice A uses $\sqrt{1225}$, which equals $\sqrt{35^2} = 35$. This suggests incorrectly adding the pool's perimeter instead of using the Pythagorean theorem. Choice B gives $25\sqrt{2}$, which would be correct if this were a square pool with sides of 25 meters, but ignores that the width is only 15 meters. Choice C has the right approach with $\sqrt{850}$ but fails to simplify the radical expression. Remember: rectangle diagonal problems always require the Pythagorean theorem, and on standardized tests, you'll often need to simplify radicals by factoring out perfect squares. Practice recognizing when $\sqrt{ab^2} = b\sqrt{a}$.

Question 9

1. 16.3° approximately (correct answer)
2. 73.7° approximately
3. 90.0° approximately
4. 106.3° approximately

Explanation: When you encounter a right triangle problem with given angle measures, remember that the three angles must always sum to 180°, and one angle is already 90°. In right triangle JKL, you know that angle K = 90° (given as the right angle) and angle J = 73.7°. Since the sum of all angles in any triangle equals 180°, you can find angle L by subtracting the known angles: 180° - 90° - 73.7° = 16.3°. You can verify this makes sense by checking the side relationships. With JL = 25 (hypotenuse) and JK = 24, angle J is the larger acute angle since it's opposite the longer leg KL. The Pythagorean theorem gives us KL = $\sqrt{25^2 - 24^2} = \sqrt{625 - 576} = 7$. Since angle L is opposite the shorter side JK = 24, it should be the smaller acute angle, which matches our calculated 16.3°. Looking at the wrong answers: A) 16.3° is actually correct. B) 73.7° incorrectly assumes angles J and L are equal, but they're not since the triangle isn't isosceles. C) 90° wrongly assigns the right angle to vertex L instead of K. D) 106.3° appears to add 90° + 16.3°, perhaps confusing interior and exterior angles, but no angle in a triangle can exceed 180°, and this would make the triangle's angle sum exceed 180°. For right triangle problems, always start with the angle sum property (180°) and remember that the two acute angles are complementary (sum to 90°).

Question 10

1. $200 exactly (correct answer)
2. $280 exactly
3. $175 exactly
4. $240 exactly

Explanation: When you see a question about finding the distance from one corner of a rectangle to the opposite corner, you're dealing with the Pythagorean theorem. The diagonal of a rectangle creates a right triangle where the sides of the rectangle are the two legs. In this problem, the garden measures 15 meters by 20 meters, so you need to find the length of the diagonal path. Using the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a = 15$ and $b = 20$. $15^2 + 20^2 = c^2$ $225 + 400 = c^2$ $625 = c^2$ $c = 25$ meters At $8 per meter, the total cost is $25 \times 8 = \200$$. Looking at the wrong answers: Answer B (280) suggests someone might have mistakenly calculated the perimeter cost ($$15 + 20 = 35$$ meters at 8 per meter). Answer C ($175) could result from calculation errors in finding the square root or in the final multiplication. Answer D ($240) might come from incorrectly using 30 meters as the diagonal length, perhaps by simply adding the two sides instead of using the Pythagorean theorem. The correct answer is A) $200 exactly. Remember this pattern: whenever you need to find the diagonal distance across a rectangle, set up a right triangle with the rectangle's dimensions as the legs. Many students forget to use the Pythagorean theorem and instead try to add or average the sides, leading to incorrect answers.

Question 11

1. 24 feet exactly (correct answer)
2. 36 feet exactly
3. 16 feet exactly
4. 28 feet exactly

Explanation: When you see a ladder leaning against a wall, you're looking at a classic right triangle problem that requires the Pythagorean theorem. The ladder forms the hypotenuse, the distance from the wall to the base of the ladder is one leg, and the height up the wall is the other leg. Using the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (ladder length) and $a$ and $b$ are the two legs. Here, you have:

• Ladder length (hypotenuse) = 26 feet
• Distance from wall (one leg) = 10 feet
• Height up wall (other leg) = unknown

Question 12

1. $4$
2. $4\sqrt{2}$
3. $4\sqrt{3}$ (correct answer)
4. $6\sqrt{2}$

Explanation: When you encounter a regular polygon inscribed in a circle, you're dealing with a problem that combines circle geometry with properties of regular polygons. The key insight is that a regular hexagon can be divided into six equilateral triangles, each with a vertex at the center. Since the hexagon is inscribed in a circle of radius 8, each side of the hexagon forms a chord of the circle. For a regular hexagon specifically, the distance from the center to each vertex equals the radius (8), and each of the six central triangles is equilateral with sides of length 8. The apothem is the distance from the center perpendicular to any side. In each equilateral triangle, this apothem represents the height from the center to the base. Using the formula for the height of an equilateral triangle with side length $s$: height = $\frac{s\sqrt{3}}{2}$. With $s = 8$, the apothem = $\frac{8\sqrt{3}}{2} = 4\sqrt{3}$. Looking at the wrong answers: (A) $4$ would be the apothem if you forgot the $\sqrt{3}$ factor entirely. (B) $4\sqrt{2}$ uses the wrong radical—this might come from confusing hexagon properties with square or octagon formulas. (D) $6\sqrt{2}$ combines incorrect coefficients and the wrong radical. The correct answer is (C) $4\sqrt{3}$. Study tip: Remember that regular hexagons create equilateral triangles when divided from the center, and equilateral triangle heights always involve $\sqrt{3}$, not $\sqrt{2}$.

Question 13

1. 62° exactly (correct answer)
2. 28° exactly
3. 90° exactly
4. 152° exactly

Explanation: When you encounter a right triangle problem with given side lengths and an angle measure, you're working with the fundamental principle that all triangles have interior angles that sum to 180°. Since triangle DEF has a right angle at F, you know one angle is 90°. The problem states that the triangle contains a 28° angle. With two angles identified (90° and 28°), you can find the third angle: $180° - 90° - 28° = 62°$. The side lengths DE = 17 and EF = 8 allow you to verify this makes sense. Since DE is opposite the right angle, it's the hypotenuse. EF = 8 is one leg, and using the Pythagorean theorem, the other leg DF would be $\sqrt{17^2 - 8^2} = \sqrt{289 - 64} = 15$. This creates a valid right triangle. Looking at the wrong answers: Choice B (28°) is incorrect because 28° is the angle already given in the problem—you're asked to find the remaining angle. Choice C (90°) is wrong because that's angle F, which is already identified as the right angle. Choice D (152°) is impossible since no interior angle in any triangle can exceed 180°, and this would make the angle sum far exceed 180°. Choice A (62°) correctly represents the third angle: $180° - 90° - 28° = 62°$. Strategy tip: In right triangle problems, immediately write down 90° as one angle, then use the angle sum property. The given side lengths often serve as verification rather than being essential to finding the angle measures.

Question 14

1. $16\sqrt{10}$ units
2. 160 units
3. $16\sqrt{5} + 8\sqrt{10}$ units (correct answer)
4. $8\sqrt{5} + 8\sqrt{10}$ units

Explanation: In an isosceles right triangle (a 45-45-90 triangle), the two legs are equal in length. Let the length of each leg be $x$. According to the Pythagorean theorem, $x^2 + x^2 = (\text{hypotenuse})^2$, which simplifies to $2x^2 = (\text{hypotenuse})^2$, so $x\sqrt{2} = \text{hypotenuse}$. We are given the hypotenuse is $8\sqrt{10}$. So, $x\sqrt{2} = 8\sqrt{10}$. To find $x$, divide by $\sqrt{2}$: $x = \frac{8\sqrt{10}}{\sqrt{2}} = 8\sqrt{\frac{10}{2}} = 8\sqrt{5}$. Each leg is $8\sqrt{5}$ units. The perimeter is the sum of the three sides: $8\sqrt{5} + 8\sqrt{5} + 8\sqrt{10} = 16\sqrt{5} + 8\sqrt{10}$ units.

Question 15

1. $6\sqrt{2}\pi$ inches
2. $12\pi$ inches (correct answer)
3. $24\pi$ inches
4. $36\pi$ inches

Explanation: First, find the side length (s) of the square from its area: $s^2 = 72$, so $s = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$ inches. The diagonal of an inscribed square is equal to the diameter of the circle. The diagonal (d) of a square is related to its side by $d = s\sqrt{2}$. So, $d = (6\sqrt{2})\sqrt{2} = 6 \times 2 = 12$ inches. The diameter of the circle is 12 inches, which means the radius (r) is 6 inches. The circumference of the circle is $C = 2\pi r = 2\pi(6) = 12\pi$ inches.

Question 16

1. $8\sqrt{2}\text{ cm}$ (correct answer)
2. $10\sqrt{2}\text{ cm}$
3. $11\sqrt{2}\text{ cm}$
4. $12\sqrt{2}\text{ cm}$

Explanation: Square side $=\dfrac{32}{4}=8$. Diagonal $=8\sqrt{2}$. Larger multiples result from misreading the perimeter.

Question 17

1. $14\text{ cm}$
2. $15\text{ cm}$ (correct answer)
3. $16\text{ cm}$
4. $17\text{ cm}$

Explanation: When you encounter a right triangle with known leg lengths, you're dealing with the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse. With legs of 9 cm and 12 cm, substitute into the formula: $9^2 + 12^2 = c^2$. This gives you $81 + 144 = c^2$, so $225 = c^2$. Taking the square root of both sides: $c = \sqrt{225} = 15$ cm. Looking at the wrong answers: Choice A (14 cm) is too small—if you squared it, you'd get 196, which is less than 225. This might result from calculation errors or forgetting to take the square root properly. Choice C (16 cm) gives you $16^2 = 256$, which exceeds our target of 225. Students sometimes pick this by rounding incorrectly or making arithmetic mistakes. Choice D (17 cm) yields $17^2 = 289$, far too large. This could happen if you added the legs instead of using the Pythagorean theorem, though even $9 + 12 = 21$ doesn't match. The answer is B (15 cm). Study tip: Memorize common Pythagorean triples like 3-4-5, 5-12-13, and 8-15-17. Notice that 9-12-15 is just the 3-4-5 triple multiplied by 3. Recognizing these patterns can save you calculation time and help you check your work quickly on the HSPT.

Question 18

1. $5\sqrt{2}\text{ in}$ (correct answer)
2. $6\sqrt{2}\text{ in}$
3. $7\sqrt{2}\text{ in}$
4. $10\sqrt{2}\text{ in}$

Explanation: When you encounter a problem involving the diagonal of a square, you're working with the Pythagorean theorem and the special properties of squares. Since all sides of a square are equal and meet at right angles, the diagonal creates two congruent right triangles. Let's call the side length $s$. When you draw a diagonal across a square, it becomes the hypotenuse of a right triangle where both legs equal $s$. Using the Pythagorean theorem: $s^2 + s^2 = \text{diagonal}^2$, which simplifies to $2s^2 = \text{diagonal}^2$. Since the diagonal is 10 inches: $2s^2 = 10^2 = 100$. Solving for $s$: $s^2 = 50$, so $s = \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ inches. Looking at the wrong answers: Choice B ($6\sqrt{2}$) might result from incorrectly using 6 instead of 5 when simplifying $\sqrt{50}$. Choice C ($7\sqrt{2}$) has no clear mathematical basis for this problem. Choice D ($10\sqrt{2}$) represents a common error where students multiply the diagonal by $\sqrt{2}$ instead of dividing—this backwards thinking assumes the diagonal is shorter than the side, which is impossible. Study tip: Remember that in any square, if the side length is $s$, the diagonal is $s\sqrt{2}$. Conversely, if the diagonal is $d$, the side length is $\frac{d}{\sqrt{2}} = \frac{d\sqrt{2}}{2}$. The diagonal is always longer than the side, never shorter.

Question 19

1. $70\text{ m}$
2. $80\text{ m}$
3. $90\text{ m}$ (correct answer)
4. $100\text{ m}$

Explanation: When you see a diagonal cutting across a rectangle, you're dealing with a right triangle where the diagonal is the hypotenuse. This is a classic Pythagorean theorem problem. The rectangular park forms a right triangle with the diagonal path. You know the length (120 m) and the diagonal (150 m), and you need to find the width. Using the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. Setting up the equation: $120^2 + w^2 = 150^2$, where $w$ is the width. Calculating: $14400 + w^2 = 22500$. Solving for $w^2$: $w^2 = 22500 - 14400 = 8100$. Therefore: $w = \sqrt{8100} = 90$ meters. Choice A (70 m) is incorrect because $120^2 + 70^2 = 14400 + 4900 = 19300$, and $\sqrt{19300} ≈ 139$ m, not 150 m. Choice B (80 m) fails because $120^2 + 80^2 = 14400 + 6400 = 20800$, giving $\sqrt{20800} ≈ 144$ m. Choice D (100 m) is wrong because $120^2 + 100^2 = 14400 + 10000 = 24400$, and $\sqrt{24400} ≈ 156$ m, which exceeds our diagonal length. Choice C (90 m) is correct because it satisfies the Pythagorean theorem perfectly. Remember: rectangular diagonal problems almost always use the Pythagorean theorem. Set up your equation carefully, identifying which measurement is the hypotenuse (usually the longest side), and double-check by substituting your answer back into the original equation.

Question 20

1. $\dfrac{3}{5}$ (correct answer)
2. $\dfrac{4}{5}$
3. $\dfrac{5}{13}$
4. $\dfrac{12}{13}$

Explanation: When you encounter a right triangle problem asking for trigonometric ratios, always start by identifying the sides relative to the angle in question. Since you need $\sin B$, you must find the ratio of the side opposite to angle $B$ divided by the hypotenuse. First, find the missing side using the Pythagorean theorem. You have $AC = 15$ (one leg) and $AB = 25$ (the hypotenuse, since it's opposite the right angle). So: $BC^2 + AC^2 = AB^2$, which gives us $BC^2 + 15^2 = 25^2$. Therefore $BC^2 = 625 - 225 = 400$, so $BC = 20$. Now for $\sin B$: the side opposite to angle $B$ is $AC = 15$, and the hypotenuse is $AB = 25$. Therefore $\sin B = \frac{15}{25} = \frac{3}{5}$. Looking at the wrong answers: Choice B ($\frac{4}{5}$) would be $\cos B$, since the adjacent side to angle $B$ is $BC = 20$, giving $\frac{20}{25} = \frac{4}{5}$. Choice C ($\frac{5}{13}$) doesn't match any ratio in this triangle and likely comes from misremembering a common Pythagorean triple. Choice D ($\frac{12}{13}$) also represents a different triangle entirely—possibly the 5-12-13 right triangle. The correct answer is A. Study tip: Always draw and label your triangle, then use SOH-CAH-TOA. For sine, you need "opposite over hypotenuse"—make sure you identify which side is opposite to your angle of interest.