Verified answer

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(i) If he hadn't apologized, he would have been punished.

(ii) It was not my intention to be rude to her.

(iii) It didn't take Mr. Swami long to adjust to the city life.

(iv) This problem proved unsolvable by anybody.

(v) We had better call the fire brigade before the fire destroys everything.

(vi) It took just two hours to repair my TV.

(vii) She said that she would play tennis next year.

(viii) There is nothing my mother doesn't know about my whereabouts.

Explanation:

Answer:

To prove that in a right-angled triangle PQR with angle PQR = 90 degrees, the mid-point M of the hypotenuse PR is equidistant from all three vertices Q, P, and R, we can use the Pythagorean theorem and the properties of midpoints.

Let's begin the proof:

Given:

1. Triangle PQR is a right-angled triangle with angle PQR = 90 degrees.

2. M is the mid-point of the hypotenuse PR.

We want to prove that QM = PM = MR.

Proof:

1. Since M is the mid-point of PR, we can write:

  PM = MR (By the definition of a midpoint).

2. Now, let's prove that QM = PM. We'll use the Pythagorean theorem.

  According to the Pythagorean theorem in right-angled triangles:

  QR² = PQ² + PR²

  Now, since we have a right-angled triangle, we can write:

  PR² = PQ² + QR² (Using the property of a right-angled triangle)

3. Now, let's consider the mid-point M of PR. By the definition of a midpoint, we can write:

  PR = 2 * PM

  Squaring both sides of this equation:

  PR² = 4 * PM²

4. Substituting the value of PR² from step 2 into step 3:

  PQ² + QR² = 4 * PM²

5. Now, let's divide both sides by 4:

  (PQ² + QR²) / 4 = PM²

6. Now, let's consider the right-angled triangle QPR. By the Pythagorean theorem:

  PQ² + QR² = PR²

7. Substitute the value of PR² from step 6 into step 5:

  (PR²) / 4 = PM²

8. Since PR is the hypotenuse of right-angled triangle PQR, and M is the midpoint of PR, PR = 2 * PM. Substitute this into step 8:

  (2 * PM)² / 4 = PM²

9. Simplify the left side of the equation:

  (4 * PM²) / 4 = PM²

10. Finally, simplify the equation:

   PM² = PM²

Since both sides of the equation are equal, we have proved that QM = PM = MR.

Hence, it is proven that in right-angled triangle PQR, the mid-point M of the hypotenuse PR is equidistant from all three vertices Q, P, and R, i.e., QM = PM = MR.

Explanation: