( $1$ ) An isosceles right triangle has an area $8\text{cm}^2$ . The length of its hypotenuse is $\newline$ $\text{cm}\sqrt{16}$ $\newline$ $\text{cm}\sqrt{48}$ $\newline$ $\text{cm}\sqrt{32}$ $\newline$ $\sqrt{24}\text{cm}$

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Grade 6

Find missing angles in special triangles

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Q. (

1

) An isosceles right triangle has an area

8\text{cm}^2

. The length of its hypotenuse is

\newline

\text{cm}\sqrt{16}

\newline

\text{cm}\sqrt{48}

\newline

\text{cm}\sqrt{32}

\newline

\sqrt{24}\text{cm}

Understand Isosceles Triangle Properties: Understand the properties of an isosceles right triangle. An isosceles right triangle has two sides of equal length, which are the legs, and the hypotenuse is the longest side. The relationship between the legs (let's call them $a$ ) and the hypotenuse (let's call it $c$ ) in such a triangle is given by the Pythagorean theorem: $a^2 + a^2 = c^2$ , which simplifies to $2a^2 = c^2$ .
Find Legs Length Using Area: Use the area to find the length of the legs. $\newline$ The area $A$ of a triangle is given by the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$ . For an isosceles right triangle, the base and height are the same and are the lengths of the legs. We know the area is $8 \, \text{cm}^2$ , so we can set up the equation: $8 = \frac{1}{2} \times a \times a$ .
Solve for 'a': Solve for 'a'. $\newline$ Multiplying both sides of the equation by $2$ to get rid of the fraction, we have: $16 = a^2$ . Taking the square root of both sides, we find that $a = \sqrt{16} = 4 \, \text{cm}$ .
Find Hypotenuse Using Legs: Use the length of the legs to find the hypotenuse. $\newline$ Now that we know the length of the legs $a = 4\,\text{cm}$ , we can use the relationship $2a^2 = c^2$ to find the hypotenuse. Plugging in the value of 'a', we get: $2 \times (4\,\text{cm})^2 = c^2$ .
Calculate Hypotenize Length: Calculate the length of the hypotenuse. $2 \times (4 \, \text{cm})^2 = 2 \times 16 \, \text{cm}^2 = 32 \, \text{cm}^2$ . So, $c^2 = 32 \, \text{cm}^2$ . Taking the square root of both sides, we find that $c = \sqrt{32} \, \text{cm}$ .
Simplify Square Root: Simplify the square root. $\sqrt{32\,\text{cm}}$ can be simplified by factoring out perfect squares: $\sqrt{32\,\text{cm}} = \sqrt{(16 \times 2)}\,\text{cm} = 4\sqrt{2}\,\text{cm}.$