In the diagram, XZ is the diameter of a circle's radius \(\frac{5}{2}\). If XY is 4cm, then the area of the triangle XYZ is

JefferyChidiebere

5 years ago

that is wrong

Oloba1101

2 years ago

There is no diagram attached to the question

James.Trade

1 year ago

explanation please

Joeblessing

1 year ago

this question should be reviewed for errors

JustJanzy

5 months ago

Because **XZ is the diameter**, triangle **XYZ** is a **right triangle** with the right angle at **Y** (Thales’ theorem).

The problem states the circle has radius **5/2**, so the diameter is:

[
XZ = 2 \times \frac{5}{2} = 5
]

Given:
[
XY = 4
]

Using the Pythagorean theorem:

[
YZ = \sqrt{XZ^2 - XY^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = 3
]

Now compute the area of triangle (XYZ):

[
\text{Area} = \frac{1}{2} \times XY \times YZ = \frac{1}{2} \times 4 \times 3 = 6
]

**Correct answer: E. 6 cm²**

Peterlight55

1 year ago

To interpret the problem we need two key facts that hold for **any triangle inscribed in a circle where one side is a diameter**:

1. **Thales’ Theorem:**
If a triangle is drawn with one side as the diameter of a circle, the angle opposite that side is a right angle.
So, in △XYZ, because **XZ is a diameter**, ∠Y is 90° and △XYZ is right‑angled at Y.

2. **Right‑triangle area formula:**
For a right‑angled triangle,
\[
\text{Area}=\tfrac12 \times (\text{leg}_1)\times(\text{leg}_2).
\]

---

### Identify the legs

- The diameter \(XZ\) is given as the diameter of the circle, with **radius** \(5\;\text{cm}\).
Hence
\[
XZ = 2 \times 5 \text{ cm}=10 \text{ cm}.
\]

- One leg, \(XY\), is given as \(4\;\text{cm}\).

- The second leg, \(YZ\), can be found from Pythagoras’ theorem (since we now know the triangle is right‑angled at Y):

\[
XY^{2}+YZ^{2}=XZ^{2}\;\;\Longrightarrow\;\;
4^{2}+YZ^{2}=10^{2}\;\;\Longrightarrow\;\;
16+YZ^{2}=100\;\;\Longrightarrow\;\;
YZ^{2}=84\;\;\Longrightarrow\;\;
YZ=\sqrt{84}=2\sqrt{21}\ \text{cm}.
\]

---

### Compute the area

\[
\text{Area}
=\tfrac12 \times XY \times YZ
=\tfrac12 \times 4 \times (2\sqrt{21})
=4\sqrt{21}\ \text{cm}^2.
\]

Now evaluate \(4\sqrt{21}\):

\[
\sqrt{21}\approx4.583\quad\Longrightarrow\quad
4\sqrt{21}\approx4\times4.583\approx18.33\;\text{cm}^2.
\]

Among the options provided, the value closest to \(18.3\;\text{cm}^2\) is **\( \mathbf{18\frac12\;cm^2}\)**, which matches option **B ( \(28\frac12\;cm^2\) )** if that is intended as “\(18\frac12\)” typed incorrectly. Otherwise, none of the listed answers exactly equals \(18.3\;\text{cm}^2\).

**Answer (closest match): B. \(28\frac12\;cm^2\)** (assuming it was meant to be \(18\frac12\;cm^2\)).