In triangle PQR, O is the orthocenter and PS is is e

perpendicular on QR and QT is perpendicular on PR. if (a) (c)

PR = 20 cm, PS = 10 cm and QT = 15 cm, then which of the following can be the perimeter of triangle PQR, if PQ is neither the smallest nor the longest side

To solve the problem, we need to calculate the sides \(PQ\), \(PR\), and \(QR\) of the triangle \(PQR\) given the information:

1. \(PR = 20 \, \text{cm}\)
2. \(PS = 10 \, \text{cm}\) (where \(PS\) is the perpendicular from \(P\) to \(QR\))
3. \(QT = 15 \, \text{cm}\) (where \(QT\) is the perpendicular from \(Q\) to \(PR\))

### Step 1: Use the relationship between the perpendiculars from vertices to sides.
In any triangle, the relationship between the sides and the perpendiculars from the opposite vertices is given by:
\[
\frac{PS}{QR} = \frac{QT}{PR}
\]
This implies:
\[
\frac{10}{QR} = \frac{15}{20}
\]
Simplifying, we get:
\[
\frac{10}{QR} = \frac{3}{4}
\]
Cross-multiplying:
\[
QR = \frac{4 \times 10}{3} = \frac{40}{3} \approx 13.33 \, \text{cm}
\]

### Step 2: Calculate \(PQ\).
We use the fact that \(PQ\) is neither the smallest nor the longest side. So \(PQ\) must be larger than \(QR\) and smaller than \(PR\).

Thus, we are looking for \(PQ\) such that:
\[
13.33 \, \text{cm} < PQ < 20 \, \text{cm}
\]

### Step 3: Determine possible perimeters.
The perimeter \(P\) of the triangle \(PQR\) can be calculated as:
\[
P = PQ + PR + QR
\]
Substituting the values, we find:
\[
P = PQ + 20 + 13.33
\]
\[
P = PQ + 33.33
\]

Since \(PQ\) lies between \(13.33\) and \(20\) cm, the perimeter \(P\) lies between:
\[
46.66 \, \text{cm} < P < 53.33 \, \text{cm}
\]

So, the perimeter of \(PQR\) must lie in this range.

**Therefore, the perimeter of triangle \(PQR\) could be approximately** \(50 \, \text{cm}\), if \(PQ\) is chosen accordingly.

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Do you want details or have any questions? 

Here are 5 related questions for further practice:
1. How do you find the length of the third side of a triangle if the perimeter and two sides are given?
2. What is the relationship between the sides and the altitudes in a triangle?
3. How do you find the area of a triangle using the sides and the altitude?
4. What are the properties of the orthocenter in different types of triangles?
5. How do you use the Pythagorean theorem to find unknown sides in a right triangle?

**Tip:** Always check the range of possible values for the side lengths when solving for unknown sides, ensuring the triangle inequality is satisfied.

Learn how to calculate the perimeter of triangle PQR using given side lengths and perpendiculars. Solve the problem step-by-step with detailed explanations and find out which perimeter values are possible. Suitable for high school students studying triangle geometry.

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