(B) Let $AB = AC = x$.In $\Delta ABC$,$m \angle A = 90^\circ$.By the Pythagorean theorem,$AB^2 + AC^2 = BC^2$.Substituting the given values: $x^2 + x^

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(B) Let $AB = AC = x$.In $\Delta ABC$,$m \angle A = 90^\circ$.By the Pythagorean theorem,$AB^2 + AC^2 = BC^2$.Substituting the given values: $x^2 + x^2 = (\sqrt{2} a)^2$.$2x^2 = 2a^2$.$x^2 = a^2$,so $x = a$ (since $a \in R^+$).Thus,$AB = AC = a$.The area of $\Delta ABC = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times AB \times AC$.Area $= \frac{1}{2} \times a \times a = \frac{1}{2} a^2$.

Class 10 Mathematics Triangles Mix Examples - Triangles

Difficult

In $\Delta ABC$,$AB = AC$ and $m \angle A = 90^\circ$. If $BC = \sqrt{2} a$,then the area of $\Delta ABC$ is $\ldots \ldots \ldots \ldots$ $(a \in R^+)$.

A
$2a$
B
$\frac{1}{2} a^2$
C
$2a^2$
D
$a^2$

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Triangles

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Mix Examples - Triangles

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $AM - CM = 10$ and $AB^{2} - BC^{2} = 260$,find $AC$.

Medium

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If $\Delta PQR \sim \Delta XYZ$ under the correspondence $PQR \leftrightarrow XYZ$,and given that $3 QR = 4 YZ$ and $PR = 8$,find the length of $XZ$.

Medium

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In $\Delta ABC$,$D$,$E$,and $F$ are the midpoints of $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ respectively. Then the correspondence $DEF \leftrightarrow \dots$ is a similarity.

Medium

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In a triangle,if the bisector of an angle bisects the side opposite to that angle,then prove that the triangle is an isosceles triangle.

Easy

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In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$,then the correspondence $ABC \leftrightarrow \dots$ is a similarity.

Easy

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In $\Delta ABC$,$AB = AC$ and $m \angle A = 90^\circ$. If $BC = \sqrt{2} a$,then the area of $\Delta ABC$ is $\ldots \ldots \ldots \ldots$ $(a \in R^+)$.

Similar Questions

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $AM - CM = 10$ and $AB^{2} - BC^{2} = 260$,find $AC$.

If $\Delta PQR \sim \Delta XYZ$ under the correspondence $PQR \leftrightarrow XYZ$,and given that $3 QR = 4 YZ$ and $PR = 8$,find the length of $XZ$.

In $\Delta ABC$,$D$,$E$,and $F$ are the midpoints of $\overline{AB}$,$\overline{BC}$,and $\overline{CA}$ respectively. Then the correspondence $DEF \leftrightarrow \dots$ is a similarity.

In a triangle,if the bisector of an angle bisects the side opposite to that angle,then prove that the triangle is an isosceles triangle.

In $\Delta ABC$ and $\Delta PQR$,if $\frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$,then the correspondence $ABC \leftrightarrow \dots$ is a similarity.

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