Answer :
To answer the given questions, we need to calculate the length of side [tex]\( AC \)[/tex] in triangle [tex]\( \triangle ABC \)[/tex] given [tex]\( AB = 14 \)[/tex] cm and [tex]\( BC = 16 \)[/tex] cm. Then, we will determine the length of [tex]\( DE \)[/tex].
Since the problem does not specify whether [tex]\( \triangle ABC \)[/tex] is a right triangle or provide any other angles, let's assume for simplicity that it is a right triangle with the right angle at [tex]\( B \)[/tex]. This assumption allows us to use the Pythagorean theorem to find the length of [tex]\( AC \)[/tex].
### Step-by-Step Solution:
#### 1.4.1 Calculate the length of [tex]\( AC \)[/tex] correct to the nearest whole number
1. Recall the Pythagorean theorem:
In a right triangle, the square of the hypotenuse [tex]\( AC \)[/tex] is equal to the sum of the squares of the other two sides [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]. This can be expressed as:
[tex]\[ AC^2 = AB^2 + BC^2 \][/tex]
2. Plug in the given values:
[tex]\[ AB = 14 \text{ cm}, \quad BC = 16 \text{ cm} \][/tex]
3. Calculate [tex]\( AB^2 \)[/tex] and [tex]\( BC^2 \)[/tex]:
[tex]\[ AB^2 = 14^2 = 196 \][/tex]
[tex]\[ BC^2 = 16^2 = 256 \][/tex]
4. Add these values to find [tex]\( AC^2 \)[/tex]:
[tex]\[ AC^2 = 196 + 256 = 452 \][/tex]
5. Take the square root to find [tex]\( AC \)[/tex]:
[tex]\[ AC = \sqrt{452} \][/tex]
6. Approximate [tex]\( \sqrt{452} \)[/tex] using a calculator:
[tex]\[ \sqrt{452} \approx 21.26 \][/tex]
7. Round to the nearest whole number:
[tex]\[ AC \approx 21 \text{ cm} \][/tex]
Thus, the length of [tex]\( AC \)[/tex] is approximately [tex]\( \boxed{21} \)[/tex] cm.
#### 1.4.2 Write down the length of [tex]\( DE \)[/tex]
The problem states that [tex]\( DE = b \)[/tex] units. It further mentions:
[tex]\[ AC = a \text{ unit} \quad \text{and } DE = b \text{ units} \][/tex]
Based on this information and the assumption made, the lengths of [tex]\( AC \)[/tex] and [tex]\( DE \)[/tex] are to be considered equal. Therefore, the length of [tex]\( DE \)[/tex] (or [tex]\( b \)[/tex]) is same as the length of [tex]\( AC \)[/tex].
Hence, the length of [tex]\( DE = b \)[/tex] is [tex]\( \boxed{21} \)[/tex] cm.
Therefore, to summarize:
- The length of [tex]\( AC \)[/tex] is [tex]\( 21 \)[/tex] cm.
- Hence, the length of [tex]\( DE \)[/tex] is also [tex]\( 21 \)[/tex] cm.
Since the problem does not specify whether [tex]\( \triangle ABC \)[/tex] is a right triangle or provide any other angles, let's assume for simplicity that it is a right triangle with the right angle at [tex]\( B \)[/tex]. This assumption allows us to use the Pythagorean theorem to find the length of [tex]\( AC \)[/tex].
### Step-by-Step Solution:
#### 1.4.1 Calculate the length of [tex]\( AC \)[/tex] correct to the nearest whole number
1. Recall the Pythagorean theorem:
In a right triangle, the square of the hypotenuse [tex]\( AC \)[/tex] is equal to the sum of the squares of the other two sides [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex]. This can be expressed as:
[tex]\[ AC^2 = AB^2 + BC^2 \][/tex]
2. Plug in the given values:
[tex]\[ AB = 14 \text{ cm}, \quad BC = 16 \text{ cm} \][/tex]
3. Calculate [tex]\( AB^2 \)[/tex] and [tex]\( BC^2 \)[/tex]:
[tex]\[ AB^2 = 14^2 = 196 \][/tex]
[tex]\[ BC^2 = 16^2 = 256 \][/tex]
4. Add these values to find [tex]\( AC^2 \)[/tex]:
[tex]\[ AC^2 = 196 + 256 = 452 \][/tex]
5. Take the square root to find [tex]\( AC \)[/tex]:
[tex]\[ AC = \sqrt{452} \][/tex]
6. Approximate [tex]\( \sqrt{452} \)[/tex] using a calculator:
[tex]\[ \sqrt{452} \approx 21.26 \][/tex]
7. Round to the nearest whole number:
[tex]\[ AC \approx 21 \text{ cm} \][/tex]
Thus, the length of [tex]\( AC \)[/tex] is approximately [tex]\( \boxed{21} \)[/tex] cm.
#### 1.4.2 Write down the length of [tex]\( DE \)[/tex]
The problem states that [tex]\( DE = b \)[/tex] units. It further mentions:
[tex]\[ AC = a \text{ unit} \quad \text{and } DE = b \text{ units} \][/tex]
Based on this information and the assumption made, the lengths of [tex]\( AC \)[/tex] and [tex]\( DE \)[/tex] are to be considered equal. Therefore, the length of [tex]\( DE \)[/tex] (or [tex]\( b \)[/tex]) is same as the length of [tex]\( AC \)[/tex].
Hence, the length of [tex]\( DE = b \)[/tex] is [tex]\( \boxed{21} \)[/tex] cm.
Therefore, to summarize:
- The length of [tex]\( AC \)[/tex] is [tex]\( 21 \)[/tex] cm.
- Hence, the length of [tex]\( DE \)[/tex] is also [tex]\( 21 \)[/tex] cm.
