At its most basic level, geometry is the mathematical exploration of shapes and space. It encompasses the study of flat, two-dimensional shapes like squares and circles, as well as three-dimensional shapes with depth, such as cubes and spheres.
Before delving into two-dimensional and three-dimensional shapes, it's crucial to understand the fundamental geometric elements that contribute to these shapes: points, lines, line segments, rays, and planes.
The concept of geometry revolves around two types of shapes: 3D and 2D shapes. Let's briefly explore them:
Q1: If angles P and Q are complementary to one another, and angle P is two times of angle Q. Calculate the measures of angles P and Q.
(a) 60°
(b) 90°
(c) 120°
(d) None
Ans: (a)
Let the measure of angle P be P, and the measure of angle Q be Q.
P = 2Q
Given that angles P and Q are complementary. Therefore,
P + Q = 90°
But, P = 2Q. Therefore,
2Q + Q = 90
3Q = 90
Q= 90/3 = 30°
P =2Q = 60°
Q2: The given triangle ABC is a right triangle. Angle ABC is 600. Find the measure of triangle APC.
(a) 56°
(b) 40°
(c) 75°
(d) 80°
Ans: (c)
The sum of all the angles in the given triangle ABC is 180°
180 = 60 + 90 + BCA
Therefore, angle BCA = 180-150= 30°
Angle APC = 60 + 45- 180
180-105 = 75°
Q3: What is the measure of angle A in the given figure?
(a) 66°
(b) 90°
(c) 85°
(d) 30°
Ans: (b)
The internal angle B in the given triangle = 180-130 = 50°
The internal angle C in the given triangle = 180-140 = 40°
Therefore, in the given triangle, measure of angle A = 180 – (40 + 50)
180 – 90 = 90°
Q4: In the given figure, determine the value of angle RPB.
(a) 60°
(b) 120°
(c) 80°
(d) 70°
Ans: (b)
In the given triangle ABC, value of angle C = 180 – (60+90)
180 – 150 = 30°
Now, in the given triangle PCR, value of angle R = 900
Value of angle CPR = 180 – (90+30)
180 – 120 = 60°
Now, value of angle RPB = (180 - 60) = 120°
Q5: What will be the area of the presented figure when all its sides are given?
(a) 11 cm²
(b) 13 cm²
(c) 10.72 cm²
(d) None of the above
Ans: (c)
The presented figure can be divided into two triangles to calculate the area.
We can also label the given figure to identify and mark the 2 triangles.
Now the figure consists of triangles: ADC and triangle ACB.
The area of the given triangle can be calculated when we know the measure of length AC.
For this, in the given triangle, ADC,
AC2 = AD2 + DC2
AC2 = 202 + 202
AC =√400+400
AC = √800
AC = 28.28
Area of triangle ADC = ½*20*20
= 200 cm²
We also need the area of the triangle ABC
Area of the triangle when its three sides are given = √p(p-a)(p-b)(p-c)
Where p = (a+b+c)/2
P = (25+5+28.28)/2 = 29.14
Therefore, area of triangle = √29.14(29.14-25)(29.14-5)(29.14-28.28)
= √115.08
= 10.72 cm²
Q6: What is the value of angle A in the given figure?
(a) 30°
(b) 70°
(c) 80°
(d) 90°
Ans: (b)
To calculate the measure of angle A, we have to calculate the value of angle B.
Angle B = 180-130= 50°
Therefore, the measure of angle A = 180 – (50+60)
A = 180-110
A =70°
Q7: In the given figure, angle CEB = 1200. Calculate the measure of angle AED.
(a) 56°
(b) 40°
(c) 120°
(d) 80°
Ans: (c)
From the given figure, angle DEB = 180-120 = 60°
Therefore, angle AED = 180-60 = 120°
Q8: Calculate the value of angle EDB when the remaining angles are given.
(a) 50°
(b) 40°
(c) 75°
(d) 90°
Ans: (a)
In the given triangle ACE, vale of angle AEC = 180- (60+40)
= 180 – 100
= 800
Also, the given angle DEB = 800 as it is directly opposite to angle AEC.
= In triangle DEB, value of angle EDB = 180 – (80+50)
=180 – 130
50°
Q9: Calculate the area enclosed by ABDEF.
(a) 1312.5 cm2
(b) 200 cm2
(c) 350 cm2
(d) None of the above
Ans: (a)
Area of ABDEF = area of ACEF – area of triangle BCD
= 50 x 30 – ((1/2)*15*25)
= 50 x 30 – 187.5
= 1500 – 187.5
= 1312.5 cm2
Q10: Determine the area of the given triangle.
(a) 25cm²
(b) 15cm²
(c) 10 cm²
(d) 55cm²
Ans: (a)
The solution is to rotate the triangle to bring the 10 feet side at the bottom.
Therefore, the area of the triangle = Area = 25cm²
1. What is the definition of geometry? |
2. What are some common geometric shapes? |
3. How is geometry used in everyday life? |
4. What are the different types of angles in geometry? |
5. What is the Pythagorean theorem in geometry? |
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