An isosceles triangle has congruent sides of $20\text{cm}$ . The base is $10\text{cm}$ . What is the area of the triangle? $\newline$ $A=81\text{cm}^2$ $\newline$ $a=9$

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Q. An isosceles triangle has congruent sides of

20\text{cm}

. The base is

10\text{cm}

. What is the area of the triangle?

\newline

A=81\text{cm}^2

\newline

a=9

Identify Triangle Area Formula: To find the area of the triangle, we need to use the formula for the area of a triangle, which is $A = \frac{1}{2} \times \text{base} \times \text{height}$ . We know the base is $10\,\text{cm}$ , but we need to find the height.
Use Pythagorean Theorem: Since the triangle is isosceles, the height will create two right-angled triangles when drawn from the vertex opposite the base to the midpoint of the base. We can use the Pythagorean theorem to find the height $h$ .
Apply Pythagorean Theorem: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse $c$ is equal to the sum of the squares of the lengths of the other two sides $a$ and $b$ . In our case, the hypotenuse is one of the congruent sides of the triangle, which is $20$ cm, and one of the other sides is half of the base, which is $5$ cm (since the height bisects the base). So we have $c = 20$ cm and $a = 5$ cm. We need to find $b$ , which is the height $h$ .
Calculate Height: Using the Pythagorean theorem, we have: $c^2 = a^2 + b^2$ , which becomes $20^2 = 5^2 + h^2$ . This simplifies to $400 = 25 + h^2$ .
Find Area Formula: Subtracting $25$ from both sides to solve for $h^2$ gives us $h^2 = 400 - 25$ , which simplifies to $h^2 = 375$ .
Calculate Area: Taking the square root of both sides to solve for $h$ gives us $h = \sqrt{375}$ . Calculating the square root of $375$ gives us $h \approx 19.36$ cm.
Calculate Area: Taking the square root of both sides to solve for $h$ gives us $h = \sqrt{375}$ . Calculating the square root of $375$ gives us $h \approx 19.36$ cm.Now that we have the height, we can find the area of the triangle using the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ . Plugging in the values, we get $A = \frac{1}{2} \times 10 \, \text{cm} \times 19.36 \, \text{cm}$ .
Calculate Area: Taking the square root of both sides to solve for $h$ gives us $h = \sqrt{375}$ . Calculating the square root of $375$ gives us $h \approx 19.36$ cm. Now that we have the height, we can find the area of the triangle using the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ . Plugging in the values, we get $A = \frac{1}{2} \times 10$ cm $\times 19.36$ cm. Calculating the area gives us $A = 5$ cm $\times 19.36$ cm, which equals $96.8$ cm $h = \sqrt{375}$ $0$ .