ISEE Upper Level Quantitative Reasoning - How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

In a right triangle, two sides have length . Give the length of the hypotenuse in terms of .

$2\sqrt{2}t$

$\sqrt{2}t$

$2\sqrt{3}t$

$\sqrt{3}t$

Explanation

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and side length.

$c^2=s^2+s^2\Rightarrow c^2=(2t)^2+(2t)^2\Rightarrow c^2=8t^2\Rightarrow c=\sqrt{8t^2}=2\sqrt{2}t$

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

$\sqrt{90}$

Explanation

To solve this problem, we are going to use the Pythagorean Theorom, which states that $a^{2}+b^{2}=c^{2}$ .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

$3^{2}+9^{2}=c^{2}$

Next, we evaluate the expoenents:

$3^{2}=9$

$9^{2}=81$

$9+81=c^{2}$

Then, $90=c^{2}$ .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply $c=\sqrt{90}$ .

Right_triangle

Refer to the above diagram, which depicts a right triangle. What is the value of ?

$3 \sqrt{ 13}$

$9\sqrt{3}$

$10\sqrt{2}$

Explanation

By the Pythagorean Theorem, which says . being the hypotenuse, or in this problem.

$x^{2} = 6^{2} + 9^{2}$

$x^{2} = 36 + 81$

$x^{2} = 117$

$x =\sqrt{ 117}$

Simply

$\sqrt{ 9} \cdot \sqrt{ 13}$

$3 \sqrt{ 13}$

The legs of a right triangle are equal to 4 and 5. What is the length of the hypotenuse?

$\sqrt{41}$

Explanation

If the legs of a right triangle are 4 and 5, to find the hypotenuse, the following equation must be used to find the hypotenuse, in which c is equal to the hypotenuse:

$a^2{+b^{2}}=c^2$

$4^2{+5^{2}}=c^2$

$41=c^{2}$

$c={\sqrt{41}}$

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

$\sqrt{65}$

$\sqrt{11}$

$\sqrt{14}$

Explanation

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

$a^{2}+b^{2}=c^{2}$

We know that the base is , so we can substitute in for . We also know that the height is , so we can substitute in for .

$7^{2}+4^{2}=c^{2}$

Next we evaluate the exponents:

$7^{2}=49$

$4^{2}=16$

Now we add them together:

Then, $65=c^{2}$ .

is not a perfect square, so we simply write the square root as $\sqrt{65}$ .

$c=\sqrt{65}$

Right_triangle

The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?

(A) The length of one side of the pentagon

(B) One and one-half feet

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

By the Pythagorean Theorem, the hypotenuse of the right triangle is

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

inches.

The pentagon in question has sides of length 75% of 112, or

$112 \times 0.75 = 84$ .

Since a pentagon has five sides of equal length, each side will have measure

$84 \div 5 = 16 \frac{4}{5}$ inches.

One and a half feet are equivalent to $12 \times 1 \frac{1}{2} = 18$ inches, so (B) is the greater quantity.

Right_triangle

The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above.

Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C.

Which is the greater quantity?

(A) The distance Cary runs

(B) One-fourth of a mile

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

By the Pythagorean Theorem, the distance from B to C is

$\sqrt{600^{2} + 800^{2} }$

$= \sqrt{360,000 + 640,000 }$

$= \sqrt{1,000,000} = 1,000$ feet

Cary runs

$800 + \frac{1}{2} \times1,000 = 800 + 500 = 1,300$ feet

Since 5,280 feet make a mile, one-fourth of a mile is equal to

$5,280 \div 4 = 1,320$ feet.

(B) is greater

In a right triangle, two sides have lengths 5 centimeters and 12 centimeters. Give the length of the hypotenuse.

$13\ cm$

$14\ cm$

$13.5\ cm$

$14.5\ cm$

$15\ cm$

Explanation

This triangle has two angles of 45 and 90 degrees, so the third angle must measure 45 degrees; this is therefore an isosceles right triangle.

By the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let hypotenuse and , lengths of the other two sides.

$c^2=a^2+b^2\Rightarrow c^2=5^2+12^2\Rightarrow c^2=25+144=169$

$\Rightarrow c=\sqrt{169}\Rightarrow c=13\ cm$

What is the hypotenuse of a right triangle with sides 5 and 8?

√89

5√4

Explanation

Because this is a right triangle, we can use the Pythagorean Theorem which says _a_2 + _b_2 = _c_2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.

_a_2 + _b_2 = _c_2

52 + 82 = _c_2

25 + 64 = _c_2

89 = _c_2

c = √89

What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?

Explanation

Since we're dealing with right triangles, we can use the Pythagorean Theorem (). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: We simplify and get . At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.

How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

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