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

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

Questions 1 - 10

Tri 3

Given the right triangle above, find the value of .

Explanation

To find the length of the side x, we must use the Pythagorean Theorem

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

which can be simplified to

Next, solve for b and we get a final answer of

$b=\sqrt{9}=3$ .

This particular example is a Pythagorean triple, or a right triangle with 3 whole number values, so it is a good one to remember.

The legs of a right triangle are $8\ cm$ and $11\ cm$ . Rounded to the nearest whole number, what is the length of the hypotenuse?

$14\ cm$

$15\ cm$

$10\ cm$

$9\ cm$

$2\ cm$

Explanation

Use the Pythagorean Theorem. The sum of both legs squared equals the hypotenuse squared.

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

(a) 55

(b)

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Explanation

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 $(AB)^{2}+ ( BC )^{2}$ and $(AC)^{2}$

$(AB)^{2}+ ( BC )^{2} = 33^{2} + 44 ^{2} = 1,089+ 1,936= 3,025$

$(AC)^{2} = 55^{2} = 3,025$

Therefore, if

$(AB)^{2}+ ( BC )^{2} =(AC)^{2}$ , 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 .

Right triangle 5

Figure NOT drawn to scale.

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

(a)

(b) 108

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

Explanation

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,

$108^{2} = 11,664$

$108^{2} >(AC) ^{2}$ , so .

Tri 5

Given the right triangle above, find the length of the missing side.

Explanation

To find the length of the side x, we must use the Pythagorean Theorem

However, this time since we are given the value of the hypotenuse, we will solve for side b rather than c.

So, when we plug the given values into the formula, the equation looks like

which can be simplified to

Next, solve for b and we get a final answer of

$b=\sqrt{1600}=40$ .

This particular example is a Pythagorean triple, or a right triangle with 3 whole number values, so it is a good one to remember.

Triangle

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

$6 \sqrt{15} \textrm{ in}$

$15 \sqrt{6} \textrm{ in}$

$3\sqrt{15} \textrm{ in}$

$12 \sqrt{ 5} \textrm{ in}$

$9 \sqrt{10} \textrm{ in}$

Explanation

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