Question

In the given figure, \(\triangle\)ABC is right angled triangle with \(\angle\)A = \(90^\circ\). AD is perpendicular to BC.

(a)(i) Prove that \(\triangle\)DBA \(\sim\) \(\triangle\)DAC

Hint:

A useful theorem to remember is: "The perpendicular drawn from the vertex of the right angle of a right triangle to the hypotenuse divides the triangle into two triangles which are similar to each other and to the original triangle."

Answer 1

Correct Answer: 4

Step 1: Understanding the Question:
This question is from the chapter Triangles, specifically dealing with the similarity of right-angled triangles when an altitude is drawn to the hypotenuse.
We are given a right-angled triangle \(ABC\) with \(\angle A = 90^\circ\) and an altitude \(AD\) perpendicular to the hypotenuse \(BC\).
We need to prove that the two smaller triangles, \(\triangle DBA\) and \(\triangle DAC\), are similar.

Step 2: Key Formula or Approach:
We will use the AA (Angle-Angle) Similarity Criterion:
If two angles of one triangle are equal to two corresponding angles of another triangle, then the two triangles are similar.

Step 3: Detailed Explanation:
1. Let us analyze the angles in the given right-angled triangle \(ABC\):
Let \(\angle B = x\).
Since \(\angle A = 90^\circ\) in \(\triangle ABC\), the sum of angles tells us: \[ \angle C = 90^\circ - x \] 2. Now, look at \(\triangle DBA\), which is right-angled at \(D\) (since \(AD \perp BC \implies \angle ADB = 90^\circ\)): \[ \angle DAB = 90^\circ - \angle B = 90^\circ - x \] 3. Now, look at \(\triangle DAC\), which is right-angled at \(D\) (since \(\angle ADC = 90^\circ\)): \[ \angle DAC = 90^\circ - \angle C = 90^\circ - (90^\circ - x) = x \] 4. Compare the angles of \(\triangle DBA\) and \(\triangle DAC\): - In \(\triangle DBA\) and \(\triangle DAC\): - \(\angle BDA = \angle ADC = 90^\circ\) (since \(AD \perp BC\))
- \(\angle ABD = \angle DAC = x\)
- \(\angle DAB = \angle ACD = 90^\circ - x\)
5. Since two corresponding angles are equal, by AA similarity: \[ \triangle DBA \sim \triangle DAC \]

Step 4: Final Answer:
Hence, it is proven that \(\triangle DBA \sim \triangle DAC\).