GAT Geometry and Spatial 3 — Questions and Answers
Question 1: A rectangle has a length of 12 cm and a width of 5 cm. What is the length of its diagonal?
- 11 cm
- 13 cm (Correct answer)
- 14 cm
- 15 cm
Correct answer: 13 cm
Using the Pythagorean theorem: diagonal = √(12² + 5²) = √(144 + 25) = √169 = 13 cm.
A rectangle's diagonal divides it into two right triangles. The diagonal is the hypotenuse, with the length and width as the two legs. By the Pythagorean theorem: d² = l² + w² = 12² + 5² = 144 + 25 = 169. Therefore d = √169 = 13 cm. This is a Pythagorean triple (5, 12, 13).
Question 2: What is the area of a circle with a diameter of 14 cm? (Use π ≈ 22/7)
- 154 cm² (Correct answer)
- 144 cm²
- 176 cm²
- 198 cm²
Correct answer: 154 cm²
Radius = 14/2 = 7 cm. Area = π × r² = (22/7) × 49 = 22 × 7 = 154 cm².
The area of a circle = πr². Given diameter = 14 cm, radius = 7 cm. Area = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 cm². Using π ≈ 22/7 is common on the GAT because it simplifies calculations with multiples of 7. With π ≈ 3.14, the answer would be approximately 153.86 cm².
Question 3: A cube has a surface area of 150 cm². What is the length of one edge?
- 4 cm
- 5 cm (Correct answer)
- 6 cm
- 7 cm
Correct answer: 5 cm
A cube has 6 identical square faces. Area of one face = 150/6 = 25 cm². Edge length = √25 = 5 cm.
Surface area of a cube = 6s² where s is the edge length. Given: 6s² = 150, so s² = 25, and s = 5 cm. The key insight is that all 6 faces of a cube are identical squares, so each face has area 150 ÷ 6 = 25 cm², and the side of each square is √25 = 5 cm.
Question 4: If a shape is rotated 90° clockwise, which of the following is true about an arrow originally pointing upward?
- It points left
- It points right (Correct answer)
- It points down
- It stays pointing up
Correct answer: It points right
A 90° clockwise rotation turns an upward-pointing arrow to point to the right.
In a 90° clockwise rotation, directions shift as follows: Up → Right, Right → Down, Down → Left, Left → Up. So an arrow pointing up will point right after a 90° clockwise rotation. This can be verified by holding a piece of paper with an arrow and rotating it. Spatial rotation questions are common on the GAT.
Question 5: Two similar triangles have a scale factor of 3:5. If the area of the smaller triangle is 27 cm², what is the area of the larger triangle?
- 45 cm²
- 75 cm² (Correct answer)
- 135 cm²
- 225 cm²
Correct answer: 75 cm²
For similar figures, the ratio of areas equals the square of the scale factor. Area ratio = (3/5)² = 9/25. So 27/x = 9/25, giving x = 75 cm².
When two figures are similar with a linear scale factor of a:b, their areas have a ratio of a²:b². Here, the linear ratio is 3:5, so the area ratio is 9:25. Setting up the proportion: 27/x = 9/25. Cross-multiply: 9x = 27 × 25 = 675. Therefore x = 75 cm². Alternatively: 27 × (25/9) = 27 × 25/9 = 3 × 25 = 75 cm².
Question 6: A right triangle has legs of length 6 cm and 8 cm. What is the perimeter of the triangle?
- 20 cm
- 24 cm (Correct answer)
- 28 cm
- 30 cm
Correct answer: 24 cm
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. Perimeter = 6 + 8 + 10 = 24 cm.
Step 1: Find the hypotenuse using the Pythagorean theorem. c² = a² + b² = 6² + 8² = 36 + 64 = 100. c = √100 = 10 cm. This is the well-known 3-4-5 Pythagorean triple scaled by 2 (6-8-10). Step 2: Perimeter = sum of all sides = 6 + 8 + 10 = 24 cm.
GAT Geometry and Spatial 3
A rectangle has a length of 12 cm and a width of 5 cm.
What is the length of its diagonal?