CBSE Class 10 Maths Chapter 6 Triangles NCERT Solutions

We are given a right-angled triangle $\triangle ABC$ where $\angle BAC = 90^{\circ}$ and $AD$ is the altitude to the hypotenuse $BC$.

Consider the triangles $\triangle ADB$ and $\triangle CDA$.

In $\triangle ADB$ and $\triangle CDA$:

1. $\angle ADB = \angle CDA = 90^{\circ}$ (since $AD \perp BC$).

2. $\angle DBA = \angle DAC$ (Angles complementary to $\angle C$ in $\triangle ABC$ and $\triangle ADC$ respectively, or by considering similarity of $\triangle ABC$ with $\triangle DBA$).

By the Angle-Angle (AA) similarity criterion, we have $\triangle ADB \sim \triangle CDA$.

Since the triangles are similar, the ratio of their corresponding sides is equal:

$\backslash frac\{BD\}\{AD\}\; =\; \backslash frac\{AD\}\{CD\}$

Cross-multiplying gives:

$BD\; \backslash cdot\; CD\; =\; AD^2$

Therefore, the correct option is (C).

Let the rhombus be $ABCD$ and its diagonals be $AC$ and $BD$. We are given the lengths of the diagonals as $AC = 16$ cm and $BD = 12$ cm.

A key property of a rhombus is that its diagonals bisect each other at right angles. Let the point of intersection of the diagonals be $O$.

Therefore, $AO = OC = \frac{1}{2} AC$ and $BO = OD = \frac{1}{2} BD$. Also, $\angle AOB = 90^{\circ}$.

Calculating the lengths of the semi-diagonals:

$AO\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 16\; \backslash text\{\; cm\}\; =\; 8\; \backslash text\{\; cm\}$

$BO\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 12\; \backslash text\{\; cm\}\; =\; 6\; \backslash text\{\; cm\}$

Now, consider the right-angled triangle $\triangle AOB$. By the Pythagorean theorem, the square of the hypotenuse ($AB$) is equal to the sum of the squares of the other two sides ($AO$ and $BO$).

$$AB^2\; =\; AO^2\; +\; OB^2$$

Substituting the values:

$AB^2\; =\; (8\; \backslash text\{\; cm\})^2\; +\; (6\; \backslash text\{\; cm\})^2$

$AB^2\; =\; 64\; \backslash text\{\; cm\}^2\; +\; 36\; \backslash text\{\; cm\}^2$

$AB^2\; =\; 100\; \backslash text\{\; cm\}^2$

Taking the square root of both sides to find the length of the side $AB$:

$AB\; =\; \backslash sqrt\{100\; \backslash text\{\; cm\}^2\}\; =\; 10\; \backslash text\{\; cm\}$

Since all sides of a rhombus are equal in length, the length of each side of the rhombus is 10 cm.

The correct option is (A).