All questions of Triangles for Class 10 Exam

Introduction
To determine the distance between the two planes after 1.5 hours, we will use the Pythagorean theorem, as they are flying at right angles to each other (one north and the other west).
Distance Calculations
- Plane 1 (North):
Speed = 1000 km/h
Time = 1.5 hours
Distance = Speed × Time = 1000 × 1.5 = 1500 km
- Plane 2 (West):
Speed = 1200 km/h
Time = 1.5 hours
Distance = Speed × Time = 1200 × 1.5 = 1800 km
Applying the Pythagorean Theorem
To find the distance between the two planes, we treat their paths as two sides of a right triangle:
- North Distance = 1500 km (Plane 1)
- West Distance = 1800 km (Plane 2)
The distance (D) between the two planes can be calculated as follows:
- D = √(North Distance² + West Distance²)
- D = √(1500² + 1800²)
Calculating the Squares
- 1500² = 2250000
- 1800² = 3240000
Now, add these values:
- Total = 2250000 + 3240000 = 5490000
Final Calculation
- D = √5490000
- D = √(1000² × 61) = 1000√61
Thus, the distance between the two planes after 1.5 hours is 300√61 km.
Conclusion
Therefore, the correct answer is option D) 300√61 km.

Given Information:
- AP = 3 cm
- PB = 6 cm
- AQ = 5 cm
- QC = 10 cm

Calculation:
- Let's denote BC as x cm.
- By using the segment addition postulate in triangle ABC, we can find the lengths of BP and PC.
- AP + PB = AB
- 3 + 6 = 9 cm
- Since AB = BC, we can substitute BC with 9 cm.
- AQ + QC = AC
- 5 + 10 = 15 cm
- Since AC = AB + BC, we can substitute AB with 9 cm and BC with x cm.
- 9 + x = 15
- x = 6 cm

Conclusion:
- BC = 6 cm
- 3PQ = 3(5) = 15 cm
- Therefore, BC = 3PQ
- Hence, the correct answer is option 'C'.

Given:
- Two isosceles triangles with equal vertical angles
- The areas of the triangles are in the ratio 225:289

To find:
- The ratio between corresponding altitudes of the triangles

Explanation:
Let's assume the two triangles are ABC and DEF, where angle BAC = angle EDF and angle ABC = angle DEF.

Step 1:
Let's assume the base of triangle ABC is BC and the base of triangle DEF is EF.
Since the triangles are isosceles, we can assume that BC = EF.

Step 2:
Let's assume the height of triangle ABC is h1 and the height of triangle DEF is h2.
We need to find the ratio h1/h2.

Step 3:
The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height.

Given that the ratio of the areas of the triangles is 225:289, we can write:
(1/2) * BC * h1 / (1/2) * EF * h2 = 225/289

Simplifying the equation, we get:
BC * h1 / EF * h2 = 225/289

Step 4:
Since BC = EF, we can eliminate them from the equation:
h1/h2 = 225/289

Step 5:
To simplify the ratio, we can divide both sides of the equation by the greatest common divisor (GCD) of 225 and 289, which is 17.

(225/17)/(289/17) = 15/17

Therefore, the ratio between the corresponding altitudes of the two triangles is 15:17.

Answer:
The ratio between the corresponding altitudes of the two triangles is 15:17. (Option B)

Explanation:

Statement a: The ratio of the areas of two similar triangles is equal to the ratio of their corresponding sides.
- This statement is incorrect. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Statement b: If a line is drawn parallel to one side of the triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.
- This statement is correct. This property is known as the Basic Proportionality Theorem (BPT) and is used to find unknown sides of similar triangles.

Statement c: All similar figures are congruent.
- This statement is incorrect. Similar figures have the same shape but not necessarily the same size, whereas congruent figures have the same shape and size.

Statement d: If in two triangles, two angles of one triangle is equal to the two angles of the other triangle then two triangles may or may not be congruent.
- This statement is correct. Two triangles with the same angle measures are not necessarily congruent, as the side lengths may differ.

Therefore, option B is the correct statement.

Given information:
- Height of the child = 3 feet
- Length of the shadow cast by the child = 2 feet
- Length of the shadow cast by the tree = 18 feet

To find: Height of the tree

The height of the tree can be determined by using the concept of similar triangles. The triangles formed by the child, its shadow, the tree, and its shadow are similar.

Similar triangles have proportional sides, meaning their corresponding sides are in the same ratio.

Using this concept, we can set up a proportion to find the height of the tree.

Let's denote:
- Height of the tree = x feet

Proportion:
(height of the tree) / (length of its shadow) = (height of the child) / (length of its shadow)

x / 18 = 3 / 2

Now, we can cross-multiply and solve for x:

2x = 18 * 3

2x = 54

x = 54 / 2

x = 27

Therefore, the height of the tree is 27 feet.

So, the correct answer is option A) 27 feet.