(A) In a right-angled triangle $\Delta ABC$ where $\angle B = 90^{\circ}$,if $\overline{BM}$ is the altitude to the hypotenuse $\overline{AC}$,then $\

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(A) In a right-angled triangle $\Delta ABC$ where $\angle B = 90^{\circ}$,if $\overline{BM}$ is the altitude to the hypotenuse $\overline{AC}$,then $\Delta AMB \sim \Delta BMC$.From the property of similar triangles,the ratio of corresponding sides is equal:$\frac{AM}{BM} = \frac{BM}{CM}$.Cross-multiplying gives $BM^2 = AM \cdot CM$.Therefore,$BM = \sqrt{AM \cdot CM}$,which means $BM$ is the geometric mean of $AM$ and $CM$.

Class 10 Mathematics Triangles Mix Examples - Triangles

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In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Then,$BM$ is the geometric mean of..............

A
$AM$ and $CM$
B
$AM$ and $AC$
C
$CM$ and $AC$
D
$AB$ and $BC$

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Mix Examples - Triangles

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$BC$ is the geometric mean of..........

Medium

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$\Delta ABC$ is an acute-angled triangle and $\overline{AM}$ is an altitude. Prove that $AC^{2} = AB^{2} + BC^{2} - 2 \cdot BC \cdot BM$.

Medium

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In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 122$ and $AD = 6$,find $BC$.

Medium

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If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,and the perimeter of $\Delta ABC$ is $45$,the perimeter of $\Delta XYZ$ is $30$,and $AB = 21$,find the length of $XZ$.

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In $\Delta GBS$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{GS}$. Prove that $\frac{GB^{2}}{BS^{2}} = \frac{GM}{SM}$.

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In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Then,$BM$ is the geometric mean of..............

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In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$BC$ is the geometric mean of..........

$\Delta ABC$ is an acute-angled triangle and $\overline{AM}$ is an altitude. Prove that $AC^{2} = AB^{2} + BC^{2} - 2 \cdot BC \cdot BM$.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 122$ and $AD = 6$,find $BC$.

If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,and the perimeter of $\Delta ABC$ is $45$,the perimeter of $\Delta XYZ$ is $30$,and $AB = 21$,find the length of $XZ$.

In $\Delta GBS$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{GS}$. Prove that $\frac{GB^{2}}{BS^{2}} = \frac{GM}{SM}$.

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