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An isosceles triangle has congruent sides of 20cm20\text{cm}. The base is 10cm10\text{cm}. What is the area of the triangle?\newlineA=81cm2A=81\text{cm}^2\newlinea=9a=9

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Q. An isosceles triangle has congruent sides of 20cm20\text{cm}. The base is 10cm10\text{cm}. What is the area of the triangle?\newlineA=81cm2A=81\text{cm}^2\newlinea=9a=9
  1. 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=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}. We know the base is 10 cm10\,\text{cm}, but we need to find the height.
  2. 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 hh.
  3. Apply Pythagorean Theorem: The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse cc is equal to the sum of the squares of the lengths of the other two sides aa and bb. In our case, the hypotenuse is one of the congruent sides of the triangle, which is 2020 cm, and one of the other sides is half of the base, which is 55 cm (since the height bisects the base). So we have c=20c = 20 cm and a=5a = 5 cm. We need to find bb, which is the height hh.
  4. Calculate Height: Using the Pythagorean theorem, we have: c2=a2+b2c^2 = a^2 + b^2, which becomes 202=52+h220^2 = 5^2 + h^2. This simplifies to 400=25+h2400 = 25 + h^2.
  5. Find Area Formula: Subtracting 2525 from both sides to solve for h2h^2 gives us h2=400−25h^2 = 400 - 25, which simplifies to h2=375h^2 = 375.
  6. Calculate Area: Taking the square root of both sides to solve for hh gives us h=375h = \sqrt{375}. Calculating the square root of 375375 gives us h≈19.36h \approx 19.36 cm.
  7. Calculate Area: Taking the square root of both sides to solve for hh gives us h=375h = \sqrt{375}. Calculating the square root of 375375 gives us h≈19.36h \approx 19.36 cm.Now that we have the height, we can find the area of the triangle using the formula A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}. Plugging in the values, we get A=12×10 cm×19.36 cmA = \frac{1}{2} \times 10 \, \text{cm} \times 19.36 \, \text{cm}.
  8. Calculate Area: Taking the square root of both sides to solve for hh gives us h=375h = \sqrt{375}. Calculating the square root of 375375 gives us h≈19.36h \approx 19.36 cm. Now that we have the height, we can find the area of the triangle using the formula A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}. Plugging in the values, we get A=12×10A = \frac{1}{2} \times 10 cm ×19.36\times 19.36 cm. Calculating the area gives us A=5A = 5 cm ×19.36\times 19.36 cm, which equals 96.896.8 cmh=375h = \sqrt{375}00.

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