Question
Ch-6: THE TRIANGLE AND ITS PROPERTIES
Worksheet No. 12
Q1. Choose the correct option:
1. In a triangle, a line segment joining any vertex to the mid-point of the opposite side is called a/an:
a) Altitude
b) Perpendicular
c) Median
d) Altitude
2. A triangle whose two angles measure \( 30^{\circ} \) and \( 120^{\circ} \) is:
a) Scalene
b) Isosceles
c) Equilateral
d) Right Angled
3. In a right-angled triangle, the lengths of two legs are 6 cm and 8 cm . The length of the hypotenuse is:
a) 14 cm
c) 10 cm
b) 11 cm
d) 12 cm
4. If \( p, q, r \) are the lengths of the three sides of a triangle, then which of the following statements are correct?
a) \( p+q=r \)
c) \( p+q>r \)
b) \( p+qr \)
5. Point of concurrence of the altitudes of a triangle is called
a) Centroid
c) Incentre
b) Orthocentre
d) Circumcentre
Q2. Fill in the blanks:
1. In an Equilateral triangle, each angle is equal to \( \qquad \) degrees.
2. A triangle cannot have more than \( \qquad \) right angle (s).
3. The longest side of a right-angled triangle is the \( \qquad \) .
4. Dimensions of a rectangle are 15 cm and 8 cm ; The length of its diagonal is \( \qquad \) cm.
5. In a triangle, an exterior angle is \( 135^{\circ} \). If the measure of one interior opposite angle is \( 45^{\circ} \) then measure of the other interior opposite angle is \( \qquad \)
Q3. State True / False:
1. An equilateral triangle is an Isosceles triangle. \( \qquad \)
2. A triangle can have sides \( 6 \mathrm{~cm}, 4 \mathrm{~cm}, 10 \mathrm{~cm} \). \( \qquad \)
3. In an isosceles triangle, two angles are equal. \( \qquad \)
4. Meeting point of the medians of a triangle is Centroid. \( \qquad \)
5. Altitudes of a triangle are always in the interior of the triangle. \( \qquad \)
Ask by Harper Bowman. in India
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
**Worksheet No. 12: THE TRIANGLE AND ITS PROPERTIES**
---
#### **Q1. Choose the correct option:**
1. **Median**
2. **Isosceles**
3. **10 cm**
4. **\( p + q > r \)**
5. **Orthocentre**
---
#### **Q2. Fill in the blanks:**
1. **60**
2. **one**
3. **hypotenuse**
4. **17**
5. **90**
---
#### **Q3. State True / False:**
1. **True**
2. **False**
3. **True**
4. **True**
5. **False**
---
**Detailed Solutions:**
1. **Median**: Connects a vertex to the midpoint of the opposite side.
2. **Isosceles**: Two equal angles imply two equal sides.
3. **Hypotenuse Calculation**: \( \sqrt{6^2 + 8^2} = 10 \text{ cm} \).
4. **Triangle Inequality**: \( p + q > r \) must hold.
5. **Orthocentre**: Intersection point of all three altitudes.
1. **60 degrees**: All angles in an equilateral triangle are equal.
2. **One**: A triangle cannot have two right angles.
3. **Hypotenuse**: Opposite the right angle.
4. **17 cm**: Diagonal \( = \sqrt{15^2 + 8^2} = 17 \text{ cm} \).
5. **90 degrees**: Exterior angle \( = 135^{\circ} \), one opposite angle \( = 45^{\circ} \), so the other is \(90^{\circ}\).
1. **True**: Equilateral triangles are isosceles.
2. **False**: Sides \(6\), \(4\), and \(10\) do not satisfy triangle inequality.
3. **True**: Isosceles triangles have two equal angles.
4. **True**: The centroid is where medians intersect.
5. **False**: In an obtuse triangle, some altitudes lie outside the triangle.
---
Feel free to ask for more help!
Solution
### **Worksheet No. 12: THE TRIANGLE AND ITS PROPERTIES**
---
#### **Q1. Choose the correct option:**
1. **In a triangle, a line segment joining any vertex to the mid-point of the opposite side is called a/an:**
**c) Median**
2. **A triangle whose two angles measure \( 30^{\circ} \) and \( 120^{\circ} \) is:**
**b) Isosceles**
3. **In a right-angled triangle, the lengths of two legs are 6 cm and 8 cm. The length of the hypotenuse is:**
**c) 10 cm**
4. **If \( p, q, r \) are the lengths of the three sides of a triangle, then which of the following statements are correct?**
**c) \( p + q > r \)**
5. **Point of concurrence of the altitudes of a triangle is called**
**b) Orthocentre**
---
#### **Q2. Fill in the blanks:**
1. **In an Equilateral triangle, each angle is equal to** **60** **degrees.**
2. **A triangle cannot have more than** **one** **right angle(s).**
3. **The longest side of a right-angled triangle is the** **hypotenuse** **.**
4. **Dimensions of a rectangle are 15 cm and 8 cm; The length of its diagonal is** **17** **cm.**
5. **In a triangle, an exterior angle is \(135^{\circ}\). If the measure of one interior opposite angle is \(45^{\circ}\) then measure of the other interior opposite angle is** **90** **degrees.**
---
#### **Q3. State True / False:**
1. **An equilateral triangle is an Isosceles triangle.** **True**
2. **A triangle can have sides \(6~\text{cm}\), \(4~\text{cm}\), \(10~\text{cm}\).** **False**
3. **In an isosceles triangle, two angles are equal.** **True**
4. **Meeting point of the medians of a triangle is Centroid.** **True**
5. **Altitudes of a triangle are always in the interior of the triangle.** **False**
---
### **Detailed Solutions:**
#### **Q1. Multiple Choice Questions**
1. **Median**: A median connects a vertex to the midpoint of the opposite side.
2. **Isosceles**: Two equal angles (\(30^{\circ}\) each) imply two equal sides.
3. **Hypotenuse Calculation**: \( \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm} \).
4. **Triangle Inequality**: The sum of any two sides must be greater than the third side.
5. **Orthocentre**: The intersection point of all three altitudes.
#### **Q2. Fill in the Blanks**
1. **60 degrees**: All angles in an equilateral triangle are equal and sum to \(180^{\circ}\).
2. **One**: A triangle cannot have two right angles as their sum would exceed \(180^{\circ}\).
3. **Hypotenuse**: The side opposite the right angle in a right-angled triangle.
4. **17 cm**: Diagonal \( = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \text{ cm} \).
5. **90 degrees**: Exterior angle \( = 135^{\circ} \), one opposite interior angle \( = 45^{\circ} \), so the other is \(135 - 45 = 90^{\circ}\).
#### **Q3. True/False Statements**
1. **True**: An equilateral triangle has all sides equal, which satisfies the condition for being isosceles.
2. **False**: The sides \(6\), \(4\), and \(10\) do not satisfy \(6 + 4 > 10\).
3. **True**: By definition, an isosceles triangle has at least two equal angles.
4. **True**: The centroid is where the three medians intersect.
5. **False**: In an obtuse triangle, some altitudes lie outside the triangle.
---
Feel free to reach out if you have any further questions or need more explanations!
Reviewed and approved by the UpStudy tutoring team
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Triangles have been a vital part of mathematics and architecture since ancient times. The Greeks, especially Euclid, laid the groundwork for triangle properties in his "Elements," which includes many foundational concepts like the Pythagorean theorem. Additionally, the properties of triangles were essential in the design of structures such as the Pyramids of Giza, showcasing their significance in both math and engineering. Triangles are not just abstract shapes; they play a crucial role in real-world applications! From engineering to art, triangles provide stability in structures such as bridges and roofs, thanks to their strong geometric properties. They are also used in computer graphics to create complex images and animations. By splitting larger shapes into triangles, designers can render visually stunning scenes in video games and movies, proving that triangles are not only fascinating in theory but also practical in daily life!
