Exploring Some Geometric Themes — Practice

Question 1

Which criterion applies when two triangles have all three sides equal?

Accepted Answer

✦ Answer: SSS (Side-Side-Side) congruence criterion.

Question 2

Find circumference of a circle with radius 14 cm (π = 22/7).

Accepted Answer

✦ Answer: C = 2πr = 2 × 22/7 × 14 = 88 cm.

Question 3

Two triangles are similar with sides in ratio 3:5. Find the ratio of their (a) perimeters, (b) areas.

Accepted Answer

Step 1 — Side ratio.
  Sides in 3:5.

Step 2 — Perimeter ratio.
  Perimeters scale linearly with sides. P₁:P₂ = 3:5.

Step 3 — Area ratio.
  Area scales as SQUARE of side ratio.
  A₁:A₂ = (3/5)² = 9/25.

✦ Answer: (a) Perimeters in 3:5. (b) Areas in 9:25.

Question 4

A wheel has diameter 70 cm. How many revolutions to cover 11 km?

Accepted Answer

Step 1 — Find circumference.
  Diameter = 70 cm, so radius = 35 cm.
  C = 2π × 35 = 70π = 70 × 22/7 = 220 cm.

Step 2 — Convert distance.
  11 km = 11,00,000 cm.

Step 3 — Number of revolutions.
  = 11,00,000 / 220 = 5,000 revolutions.

✦ Answer: 5,000 revolutions.

Question 5

A 6 m tall man casts a shadow of 3 m. At the same time, a building casts a shadow of 45 m. Using similar triangles, find the height of the building.

Accepted Answer

Step 1 — Identify similar triangles.
  Sun's rays make the same angle with the ground.
  Triangle 1: man (6 m), shadow (3 m), sun ray (hypotenuse)
  Triangle 2: building (h m), shadow (45 m), sun ray
  These are SIMILAR (AA similarity — all triangles have a right angle + same sun angle).

Step 2 — Set up proportion.
  Height : Shadow ratio is constant.
  h / 45 = 6 / 3

Step 3 — Solve.
  h = 45 × 6 / 3 = 90 m

Step 4 — Verify.
  Man: 6/3 = 2 (height-to-shadow ratio)
  Building: 90/45 = 2 ✓ (same ratio)

✦ Answer: The building is 90 m tall. (This is the classic 'shadow method' used by ancient surveyors!)

Practice — Exploring Some Geometric Themes

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