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1 Hypotenuse of a triangle is '
b '. Find the maximum area of the triangle.

$1$ Hypotenuse of a triangle is ' $b$ '. Find the maximum area of the triangle.

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Area of quadrilaterals and triangles: word problems

Full solution

Q.

1

Hypotenuse of a triangle is '

b

'. Find the maximum area of the triangle.

Triangle Construction: To find the maximum area of a triangle with a given hypotenuse, we can use a right-angled triangle where the two legs are equal, because this will give us the maximum area. The legs will each be $\frac{b}{\sqrt{2}}$ long. $\newline$ Calculation: Legs = $\frac{b}{\sqrt{2}}$
Area Calculation: Now, we use the formula for the area of a triangle, which is $\frac{1}{2} \times \text{base} \times \text{height}$ . Since our triangle is right-angled and the legs are equal, the base and height are both ' $\frac{b}{\sqrt{2}}$ '. $\newline$ Calculation: Area = $\frac{1}{2} \times \left(\frac{b}{\sqrt{2}}\right) \times \left(\frac{b}{\sqrt{2}}\right)$
Area Simplification: Simplify the area formula by multiplying the base and height and then multiplying by $\frac{1}{2}$ . $\newline$ Calculation: Area = $\frac{1}{2} \times \left(\frac{b^2}{2}\right)$

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