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

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.

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.

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

$12\ cm$

$10\ cm$

$6\ cm$

$14\ cm$

$7\ cm$

Explanation

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

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

where is the hypotenuse and and are the sides.

$\sqrt{(15^{2}-9^{2})} = 12$

Find the length of the missing side.

Explanation

Recall the Pythagorean Theorem for a right triangle:

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

$b=\sqrt{c^2-a^2}$

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

$b=\sqrt{22^2-p^2}=\sqrt{403}=20.07$

The three sides of a triangle have lengths 0.8, 1.2, and 1.5.

True or false: the triangle is a right triangle.

False

True

Explanation

By the Pythagorean Theorem and its converse, a triangle is right if and only if

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

where is the length of the longest side and and are the lengths of the other two sides.

Therefore, set and test the statement for truth or falsity:

$0.8^{2}+1.2^{2}= 1.5^{2}$

The statement is false, so the Pythagorean relationship does not hold. The triangle is not right.

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

Explanation

We use the Pythagorean Theorem so the problem to solve becomes $10^{2}+x^{2}=26^{2}$ where = unknown leg length

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

The sum of the two legs becomes

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ .

$\angle B$ is an acute angle; $\angle E$ is a right angle.

$\overline{AB} \cong \overline{DE}$

$\overline{BC} \cong \overline{EF}$

Which is a true statement?

Explanation

$\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$ . However, the included angle of $\overline{AB}$ and $\overline{BC}$ , $\angle B$ , is acute, so its measure is less than that of $\angle E$ , which is right. This sets up the conditions of the SAS Inequality Theorem (or Hinge Theorem); the side of lesser length is opposite the angle of lesser measure. Consequently, .