Here are the step-by-step solutions for the three problems on the worksheet.
Q1) Isosceles Triangle
Problem: Calculate the perpendicular distance from A to BC in an isosceles triangle where $AB = AC = 12$ cm and $BC = 10$ cm.
Step-by-Step Solution:
1.
Understand the Geometry: In an isosceles triangle, if you draw a line straight down from the top corner (A) to the base (BC), it cuts the base exactly in half. It also creates two right-angled triangles.
2.
Find the Base of the Right Triangle: Since the total base $BC$ is $10$ cm, half of the base is:
$$10 \div 2 = 5 \text{ cm}$$
3.
Identify the Sides: Now look at one of the right-angled triangles formed:
* The hypotenuse (the long slanted side) is side $AB$, which is
12 cm.
* The bottom side is half of $BC$, which is
5 cm.
* The vertical side is the height we need to find. Let's call it $h$.
4.
Use Pythagoras' Theorem: The formula is $a^2 + b^2 = c^2$.
$$h^2 + 5^2 = 12^2$$
5.
Calculate Squares:
$$h^2 + 25 = 144$$
6.
Solve for $h$: Subtract 25 from 144.
$$h^2 = 144 - 25$$
$$h^2 = 119$$
7.
Final Calculation: Take the square root of 119.
$$h = \sqrt{119} \approx 10.91 \text{ cm}$$
***
Q2) Length of AB using Coordinates
Problem: Work out the length of AB when coordinates of A are $(4, 7)$ and B are $(16, 12)$.
Step-by-Step Solution:
1.
Find the Horizontal Distance: Look at the x-coordinates (the first numbers).
$$16 - 4 = 12$$
2.
Find the Vertical Distance: Look at the y-coordinates (the second numbers).
$$12 - 7 = 5$$
3.
Visualize a Triangle: These distances form the two shorter sides of a right-angled triangle. The line AB is the hypotenuse.
4.
Use Pythagoras' Theorem:
$$\text{Length}^2 = 12^2 + 5^2$$
5.
Calculate Squares:
$$\text{Length}^2 = 144 + 25$$
$$\text{Length}^2 = 169$$
6.
Final Calculation: Take the square root of 169.
$$\text{Length} = \sqrt{169} = 13$$
***
Q3) Ladder Problem
Problem: A ladder 8 metres long leans against a wall. The bottom is 6 metres from the wall. Calculate the height of the windowsill.
Step-by-Step Solution:
1.
Identify the Sides: This forms a right-angled triangle.
* The ladder is the hypotenuse ($c$) =
8 m.
* The distance from the wall is the base ($b$) =
6 m.
* The height of the wall is the side we need to find ($a$).
2.
Use Pythagoras' Theorem:
$$a^2 + 6^2 = 8^2$$
3.
Calculate Squares:
$$a^2 + 36 = 64$$
4.
Solve for $a$: Subtract 36 from 64.
$$a^2 = 64 - 36$$
$$a^2 = 28$$
5.
Final Calculation: Take the square root of 28.
$$a = \sqrt{28} \approx 5.29 \text{ m}$$
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Final Answer:
Q1) $\sqrt{119}$ cm or approx
10.91 cm
Q2) 13
Q3) $\sqrt{28}$ m or approx
5.29 m
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem worksheet word problems.
