Sure, I’d be happy to help! Let's go through each question step-by-step.
### 7. The sides of the triangle are [tex]\(35 \, \text{cm}\)[/tex], [tex]\(54 \, \text{cm}\)[/tex], and [tex]\(61 \, \text{cm}\)[/tex]. Find the length of its longest altitude.
First, let's identify the given sides of the triangle:
[tex]\[ a = 35 \, \text{cm}, \, b = 54 \, \text{cm}, \, c = 61 \, \text{cm} \][/tex]
Assuming [tex]\(c\)[/tex] is the longest side, the longest altitude will drop to side [tex]\(c\)[/tex].
We can find the area [tex]\(A\)[/tex] of the triangle using Heron's formula.
Heron's formula is given by:
[tex]\[ s = \frac{a + b + c}{2} \, \text{(semi-perimeter)} \][/tex]
[tex]\[ s = \frac{35 + 54 + 61}{2} \][/tex]
[tex]\[ s = 75 \, \text{cm} \][/tex]
Next, we calculate the area [tex]\(A\)[/tex] of the triangle:
[tex]\[ A = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
[tex]\[ A = \sqrt{75 (75 - 35) (75 - 54) (75 - 61)} \][/tex]
[tex]\[ A = \sqrt{75 \times 40 \times 21 \times 14} \][/tex]
[tex]\[ A = 939.1485505499116 \, \text{cm}^2 \][/tex]
Now, we use the formula for the altitude [tex]\(h\)[/tex] to side [tex]\(c\)[/tex]:
[tex]\[ h = \frac{2A}{c} \][/tex]
[tex]\[ h = \frac{2 \times 939.1485505499116}{61} \][/tex]
[tex]\[ h = 30.79175575573481 \, \text{cm} \][/tex]
So, the length of the longest altitude is approximately [tex]\(30.79 \, \text{cm}\)[/tex].
### 8. Factorise the following expressions.
#### i) [tex]\( x^2 + 11x + 30 \)[/tex]
To factor this quadratic expression, we look for two numbers that multiply to 30 (the constant term) and add to 11 (the coefficient of the linear term).
These numbers are 5 and 6:
[tex]\[ x^2 + 11x + 30 = (x + 5)(x + 6) \][/tex]
#### ii) [tex]\( x^2 - 2\sqrt{2}x - 30 \)[/tex]
Similarly, we factor this quadratic expression by looking for two numbers that multiply to [tex]\(-30\)[/tex] and add to [tex]\(-2\sqrt{2}\)[/tex].
These numbers are [tex]\((\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{5})\)[/tex]:
[tex]\[ x^2 - 2\sqrt{2}x - 30 = \left(x - (\sqrt{2} + \sqrt{3})\right)\left(x + (\sqrt{2} - \sqrt{3})\right)\][/tex]
### 9. Expand.
Please provide the expression to expand, and I will guide you through the steps to expanding it correctly.
I hope this helps clarify the steps for solving these problems. Let me know if there's anything else you need!
