Right Triangles MCQ Quiz in मल्याळम - Objective Question with Answer for Right Triangles - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

In a right triangle, the tangent of one acute angle is the reciprocal of the tangent of the other. Given \( \text{tan} \angle A = \frac{1}{3} \), the reciprocal is \( 3 \). Therefore, \( \text{tan} \angle B = 3 \). This reciprocal relationship holds because the sum of \( \angle A \) and \( \angle B \) must be \( 90^\circ \).

The diagonal \(d\) of a square with side length \(s\) is given by the formula \(d = s\sqrt{2}\). For a square with side length \(15\) inches, the diagonal is \(15\sqrt{2}\) inches. Thus, option 4 is correct. Option 1 is incorrect as it provides the side length, not the diagonal. Option 2 is incorrect as it erroneously doubles the side length. Option 3 is incorrect as it represents a miscalculation.

Using the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, we substitute \(a = 9\) and \(c = 15\). We have \(9^2 + b^2 = 15^2\), which is \(81 + b^2 = 225\). Solving for \(b^2\), \(b^2 = 144\), so \(b = 12\) inches. Option 2 is correct. Option 1 is incorrect as it represents a miscalculation. Option 3 is incorrect because it provides the length of the given leg. Option 4 is incorrect due to calculation errors.

In a right triangle with a \(30^\circ\) angle, the side opposite the \(30^\circ\) angle is half the hypotenuse. Since the hypotenuse is \(10\) units, the opposite side is \(\frac{10}{2} = 5\) units. Thus, the correct answer is option 3, \(5\) units.

In a \(30-60-90\) triangle, the side opposite the \(30^\circ\) angle is half the hypotenuse. Therefore, the length of the side opposite the \(30^\circ\) angle is \(\frac{16}{2} = 8\) inches. Option 1 is correct. Option 2 is incorrect because it mistakenly suggests the opposite side is equal to the hypotenuse. Option 3 is incorrect as it is the length of the side opposite the \(60^\circ\) angle. Option 4 is incorrect as it represents a miscalculation.

In a rectangle, the diagonal forms the hypotenuse of a right triangle with the length and width. Using the Pythagorean theorem, \(9^2 + l^2 = 15^2\), where \(l\) is the length. This becomes \(81 + l^2 = 225\). Subtract \(81\) from both sides to get \(l^2 = 144\). Taking the square root gives \(l = 12\). Thus, the length of the garden is \(12\) meters.

The Pythagorean theorem states that in a right triangle, \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. Here, \(a = 6\) and \(c = 10\). We need to find \(b\). Substituting the known values, we have \(6^2 + b^2 = 10^2\). This simplifies to \(36 + b^2 = 100\). Solving for \(b^2\), we subtract \(36\) from both sides, yielding \(b^2 = 64\). Taking the square root of both sides gives \(b = 8\). Therefore, the length of the other leg is \(8\).

The diagonal forms a right triangle with the width and length of the pool. Using the Pythagorean theorem, \(7^2 + l^2 = 25^2\), where \(l\) is the length. This simplifies to \(49 + l^2 = 625\). Subtract \(49\) from both sides to get \(l^2 = 576\). Taking the square root gives \(l = 24\). Therefore, the length of the pool is \(24\) meters.

The ratio of the sides is \(3:4:5\), which is a common Pythagorean triplet. Let \(3x\), \(4x\), and \(5x\) be the sides. The longest side \(5x = 20\). Solving for \(x\), we get \(x = 4\). The shortest side is \(3x = 3 \times 4 = 12\). Therefore, the length of the shortest side is \(12\) cm.

The ladder forms a right triangle with the wall and the ground. Using the Pythagorean theorem, where the height \(12\) feet and the ladder \(13\) feet are given, we need to find the base \(b\). The equation is \(b^2 + 12^2 = 13^2\). Simplifying, we have \(b^2 + 144 = 169\). Subtract \(144\) from both sides to get \(b^2 = 25\). Taking the square root gives \(b = 5\). Hence, the base of the ladder is \(5\) feet from the wall.