1) Plot and construct the triangle with the coordinates of the vertices of RST are $$ R(-2,-3), S(8,2) $$, and $$ T(4,5) $$. 1. Which type of triangle is RST 11. Which side of the triangle is the longest ? Sedadon III. State and measure side ST (SB) Tongest? BS What is the perimeter of RST? 30 (4marks)

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UpStudy AI · Accepted Answer

**Step 1. Calculate the side lengths using the distance formula**

For vertices  
$$
R(-2,-3),\quad S(8,2),\quad T(4,5),
$$  
the distance between two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$ is  
$$
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
$$

1. Side $$RS$$ (between $$R$$ and $$S$$):
   $$
   RS=\sqrt{(8-(-2))^2+(2-(-3))^2}=\sqrt{(10)^2+(5)^2}=\sqrt{100+25}=\sqrt{125}=5\sqrt{5}.
   $$

2. Side $$ST$$ (between $$S$$ and $$T$$):
   $$
   ST=\sqrt{(4-8)^2+(5-2)^2}=\sqrt{(-4)^2+(3)^2}=\sqrt{16+9}=\sqrt{25}=5.
   $$

3. Side $$TR$$ (between $$T$$ and $$R$$):
   $$
   TR=\sqrt{(-2-4)^2+(-3-5)^2}=\sqrt{(-6)^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=10.
   $$

---

**Step 2. Determine the Type of Triangle**

The side lengths are:
- $$ RS=5\sqrt{5} $$  
- $$ ST=5 $$  
- $$ TR=10 $$

Check if the triangle is right-angled using the Pythagorean theorem. The longest side is $$ RS=5\sqrt{5} $$ (since $$5\sqrt{5}\approx11.18$$), so verify:
$$
ST^2+TR^2 = 5^2+10^2 = 25+100 = 125,
$$
and
$$
RS^2=(5\sqrt{5})^2=25\times5=125.
$$

Since
$$
ST^2+TR^2=RS^2,
$$
the triangle is right-angled with the right angle at vertex $$T$$ (the vertex opposite the longest side $$RS$$). Moreover, all side lengths are different, so triangle RST is a scalene right triangle.

---

**Step 3. Identify the Longest Side**

The longest side is $$RS=5\sqrt{5}$$ (approximately $$11.18$$).

---

**Step 4. State and Measure Side $$ST$$ and Find the Perimeter**

- The length of side $$ST$$ is $$5$$.

- The perimeter of triangle RST is the sum of its side lengths:
  $$
  \text{Perimeter} = RS + ST + TR = 5\sqrt{5} + 5 + 10 = 15 + 5\sqrt{5}.
  $$

---

**Final Answers**

- **Type of Triangle:** Triangle RST is a scalene right triangle (right-angled at $$T$$).
- **Longest Side:** $$RS=5\sqrt{5}$$.
- **Measure of Side $$ST$$:** $$ST=5$$.
- **Perimeter of Triangle RST:** $$15+5\sqrt{5}$$.
**Triangle RST is a scalene right triangle with the right angle at vertex T. The longest side is RS, which is approximately 11.18 units long. Side ST measures 5 units. The perimeter of triangle RST is $$15 + 5\sqrt{5}$$ units.**

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