It is given that a banner shaped like a right triangle has a hypotenuse of $26$ feet and a leg of $10$ feet. We will find the area of the banner.

We will first identify what we need to find the area of a triangle. Let's remember that the area of a triangle is half the product of the base

b

and the height

h .

Therefore, to find the area of a triangle, we need the length of a base and its corresponding height.

A = \frac{1}{2} b h

Notice that in our right triangle, when we consider the given leg of

10

feet as a base, the other leg is the corresponding height

h .

Therefore, in order to calculate the area of our triangle, we need to find the length of the other leg

h .

Since the right triangle has a hypotenuse of

26

feet and a leg of

10

feet, we can use the Pythagorean Theorem to find the value of

h .

a^{2} + b^{2} = c^{2} \Leftrightarrow 10^{2} + h^{2} = 26^{2}

Let's solve this equation for

h .

1 0^{2} + h^{2} = 2 6^{2}

Calculate power

100 + h^{2} = 676

$LHS - 100 = RHS - 100$

h^{2} = 576

$LHS = RHS$

h = 576

Calculate root

h = 24

The length of the other leg is

24

feet. Now, we have all the information needed to find the area of the banner. Let's substitute

b = 10

and

h = 24

into the area formula, and calculate it.

A = \frac{1}{2} b h

$b = 10$ , $h = 24$

A = \frac{1}{2} (10) (24)

A = \frac{1}{2} (240)

$\frac{1}{b} \cdot a = \frac{a}{b}$

A = \frac{2 4 0}{2}

Calculate quotient

A = 120

We found that the area of the banner is

120 feet^{2} .

Exercise