# Precalculus : Solve a Right Triangle

## Example Questions

### Example Question #1 : Solve A Right Triangle

In a right triangle, if the hypotenuse is  and a leg is , what is the area of the triangle?

Explanation:

Use the Pythagorean Theorem to find the other leg.

The length of the given leg is 3, and the unknown leg is .

Use the area of a triangle formula and solve.

### Example Question #2 : Solve A Right Triangle

An isosceles right triangle has a hypotenuse of 1.  What is the area of this triangle?

Explanation:

Write the formula for the Pythagorean theorem.

In an isosceles right triangle, both legs of the right triangle are equal.

Substitute the either variable and the known hypotheuse and determine the side length.

This length represents both the base and the height of the triangle. Write the area of a triangle and substitute to solve for the area.

### Example Question #3 : Solve A Right Triangle

Solve the right triangle.

C=90°

B=45°

a=5

c=

A=135°

b=5

A=135°

b=2.07

A=45°

b=5

None of these answers are correct.

A=45°

b=

A=45°

b=5

Explanation:

Given that:

C=90°

B=45°

a=5

c=

Therefore...

All angles of a triangle add up to 180°.

### Example Question #4 : Solve A Right Triangle

A right triangle has a base of 10 and a hypotenuse of 20.  What is the length of the other leg?

Explanation:

Write the Pythagorean Theorem.

Substitute the values of the leg and hypotenuse.  The hypotenuse is the longest side of the right triangle.  Solve for the unknown variable.

### Example Question #1 : Trigonometric Applications

In the right triangle ABC, side AB is  cm long, side AC is  cm long, and side BC is the hypotenuse. How long is side BC?

cm

cm

cm

cm

cm

Explanation:

Given that ABC is a right triangle, the length of hypotenuse BC is the root of the sum of the squares of the two other sides (in other words, . Since AB is  cm long and AC is  cm long, we get that , and so .

### Example Question #6 : Solve A Right Triangle

The side lengths of right triangle ABC are such that AC > BC > AB. AC = 25 and AB = 9. What is the length of BC?

Explanation:

When you are using Pythagorean Theorem to calculate the missing side of a right triangle, it is crucial that you identify which side is the hypotenuse,  in the Pythagorean equation . Here you're told that side AC is the longest of the three sides, so 25 will serve as the length of the hypotenuse and the value of . This allows you to set up the equation:

And then you can perform the calculations on the known values:

Meaning that:

From there you can simplify, arriving at a = 4 times the square root of 34.

### Example Question #7 : Solve A Right Triangle

Given , and the lower angles of the isosceles triangle are , what is the length of ? Round to the nearest tenth.

Explanation:

Since the angle of the isosceles is , the larger angle of the right triangle formed by  is also .

Using , we can find

.

Then solve for

.

Simplify: .

Lastly, round and add appropriate units: .

### Example Question #8 : Solve A Right Triangle

In isosceles triangle , .  If side , what is the approximate length of the two legs  and ?

Explanation:

In the diagram, AB is cut in half by the altitude.

From here it easy to use right triangle trigonometry to solve for AC.

### Example Question #9 : Solve A Right Triangle

Find the area of the following Isosceles triangle (units are in cm):

Explanation:

The formula for the area of a triangle is:

We already know what the base is and we can find the height by dividing the isosceles triangle into 2 right triangles:

From there, we can use the Pathegorean Theorem to calculate height:

To find the area, now we just plug these values into the formula:

### Example Question #10 : Solve A Right Triangle

Find the area of the given isosceles triangle and round all values to the nearest tenth: