How to find the length of the side of a right triangle - Math

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Question

Refer to the above right triangle. Which of the following is equal to ?

Answer

By the Pythagorean Theorem,

$x = $\sqrt{180}$ = $\sqrt{36}$ \cdot $\sqrt{5}$ = 6 $\sqrt{5}$$

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Question

Given $\Delta ABC$ with right angle $\angle B$ , $m \angle C = 45 ^{\circ }$

Which is the greater quantity?

(a)

(b)

Answer

$m \angle B = $90^{\circ}$$

$m \angle C = 45 ^{\circ }$

The sum of the measures of the angles of a triangle is , so:

$m \angle A + m \angle B + m \angle C = 180$

$m \angle A +90 + 45 = 180$

$m \angle A +135= 180$

$m \angle A +135-135= 180-135$

$m \angle A = $45^{\circ }$$

This is a triangle, so its legs $\overline{AB}$ and $\overline{BC}$ are congruent. The quantities are equal.

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Question

Give the length of one leg of an isosceles right triangle whose area is the same as the right triangle in the above diagram.

Answer

The area of a triangle is half the product of its height and its base; in a right triangle, the legs, being perpendicular, can serve as these quantites.

The triangle in the diagram has area

$$\frac{1}{2}$ \times 30 \times 18 = 270$ square inches.

An isosceles right triangle has two legs of the same length, which we will call . The area of that triangle, which is the same as that of the one in the diagram, is therefore

$$\frac{1}{2}$ L ^{2} = 270$

$$\frac{1}{2}$ L ^{2}\cdot 2 = 270 \cdot 2$

$L ^{2} = 540$

$L = $\sqrt{540}$$

$L = $\sqrt{36}$\cdot $\sqrt{15}$ = 6 $\sqrt{15}$$ inches.

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Question

The perimeter of a regular octagon is 20% greater than that of the above right triangle. Which is the greater quantity?

(A) The length of one side of the octagon

(B) 3 yards

Answer

By the Pythagorean Theorem, the shorter leg has length

feet.

The perimeter of the right triangle is therefore

feet.

The octagon has perimeter 20% greater than this, or

$60 + 0.20 \times 60 = 60 + 12 = 72$ feet.

A regular octagon has eight sides of equal length, so each side of this octagon has length

$72 \div 8 = 9$ feet, which is equal to 3 yards. This makes the quantities equal.

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Question

The area of a square is equal to that of the above right triangle. Which is the greater quantity?

(A) The sidelength of the square

(B) 4 yards

Answer

By the Pythagorean Theorem, the shorter leg has length

feet.

The area of a triangle is equal to half the product of its base and height; for a right triangle, the legs can serve as these. The area of the above right triangle is

$$\frac{1}{2}$ \times 10 \times 24 = 120$ square feet.

The sidelength is the square root of this; , so $$\sqrt{120}$ < $\sqrt{121}$ = 11$ . Therefore each sidelength of the square is just under 11 feet. 4 yards is 12 feet, so (B) is greater.

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Question

Consider a triangle, $\bigtriangleup ABC$ , in which , , and $m \angle B = 100 ^{\circ}$ . Which is the greater quantity?

(a) 55

(b)

Answer

Suppose .

By the Converse of the Pythagorean Theorem, a triangle is right if and only if the sum of the squares of the lengths of the smallest two sides is equal to the square of the longest side. Compare the quantities and

Therefore, if

, so $\bigtriangleup ABC$ is right, with the right angle opposite longest side $\overline{AC}$ . Thus, $\angle B$ is right and has degree measure 90.

However, $\angle B$ has degree measure greater than 90, so, as a consequence of the Converse of the Pythagorean Theorem and the SAS Inequality Theorem, it holds that .

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Question

Figure NOT drawn to scale.

Refer to the above triangle. Which is the greater quantity?

(a)

(b) 108

Answer

We can compare these numbers by comparing their squares.

By the Pythagorean Theorem,

$(AC) ^{2}= (AB) ^{2}+ $(BC)^{2}$ = 100 ^{2}+ 40 ^{2} = 10,000 + 1,600 = 11,600$

Also,

, so .

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Question

Find the length of the missing side length.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

What is the length of the remaining side of the right triangle?

Answer

Rearrange the Pythagorean Theorem to find the missing side. The Pythagorean Theorem is:

where is the hypotenuse and and are the sides.

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Question

Find the length of the unknown side, , of the right triangle below.

Answer

We need to use the Pythagorean Theorem, which says that .

Since we need to find the length of 'a', we can just solve for 'a'.

In our case, c = 12 and b = 9.

Thus, .

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Question

The hypotenuse of a right triangle is 26 in and one leg is 10 in. What is the sum of the two shortest sides?

Answer

We use the Pythagorean Theorem so the problem to solve becomes where = unknown leg length

So $100 + $x^{2}$=676$ and

The sum of the two legs becomes

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Question

$In;\Delta KLM,;what;is;the;length;of;$\overline{KL}$?$

Answer

$\fn_cm $A^2$$+4^2$$=5^2$$

$A=\sqrt9$

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Question

Find the missing leg of the right triangle when one of the legs is and the hypotenuse is .

Answer

In order to find the missing side of a right triangle you must use one of two things:

1. Pythagorean Theorem

2. Trigonometry.

Since we only know what the side lengths are we must use the Pythagorean Theorem.

a=4, b=x, and c=5

So the missing side length is 3

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Question

Given that the hypotenuse of a right triangle is and one side is , find the other side length.

Answer

To find the length of the missing side, you can either use the pythagorean theorm or realize this is a case of a special right triangle with sides .

Thus, the other side length is .

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

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Question

Find the length of the missing side.

Answer

Recall the Pythagorean Theorem for a right triangle:

Since the missing side corresponds to side , rewrite the Pythagorean Theorem and solve for .

Now, plug in values of and into a calculator to find the length of side . Round to decimal places.

Compare your answer with the correct one above

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