If ABC is an isosceles triangle with ∠C=90° and AC=5 cm; then AB is? The straight line given by the equation x=11 is: a) parallel to X axis b) parallel to Y axis c) passes through... If ABC is an isosceles triangle with ∠C=90° and AC=5 cm; then AB is? The straight line given by the equation x=11 is: a) parallel to X axis b) parallel to Y axis c) passes through the origin d) passes through the point (0, 11). If two non-vertical lines are perpendicular if and only if: a) m1=m2 b) m1=m2 c) m1m2=-1 d) m1m2=1. If the ratio of the height of a tower and the length of its shadow is √3:1, then the angle of elevation of the sun has measure: a) 45° b) 30° c) 90° d) 60°. Read more

Steps to Solve

1. Understanding the Isosceles Triangle Problem

For triangle ( \triangle ABC ), where ( \angle C = 90^\circ ) and ( AC = 5 , \text{cm} ), we have an isosceles triangle. In such a triangle, the two legs are equal. Let ( AB ) be ( x ). Since ( \triangle ABC ) has ( AC = BC = 5 , \text{cm} ):

$$ x = AB = \sqrt{AC^2 + BC^2} = \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2} , \text{cm} $$

2. Evaluating the Equation of the Line

The equation ( x = 11 ) represents a vertical line, which means it is:

• Not parallel to the X-axis, since vertical lines are perpendicular to horizontal lines.
• Not parallel to the Y-axis, as it is indeed a vertical line.
• Does not pass through the origin (0, 0) because it passes through (11, y).
• Passes through the point (11, y), thus confirming that it passes through points where x = 11.

3. Perpendicular Lines Condition

Non-vertical lines are perpendicular if:

$$ m_1 m_2 = -1 $$

This means that the product of their slopes should equal -1.

4. Calculating the Angle of Elevation

Using the ratio of the height of the tower to the length of its shadow (height: shadow):

Let height = ( h ) and shadow = ( s ):

$$ \tan(\theta) = \frac{h}{s} $$

Given ( \frac{h}{s} = \sqrt{3} : 1):

Thus,

$$ \tan(\theta) = \sqrt{3} $$

This corresponds to:

$$ \theta = 60^\circ $$