Question

If \(\mathrm{P}_{1}, \mathrm{P}_{2}, \mathrm{P}_{3}\) are the perpendiculars drawn from \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) to the opposite sides of the triangle \(\mathrm{ABC}\), then \(\mathrm{P}_{1} \mathrm{P}_{2} \mathrm{P}_{3}\) is (a) \(\frac{2 \Delta}{\mathrm{R}}\) (b) \(\frac{2 \Delta}{\mathrm{r}}\) (c) \(\frac{2 \Delta^{2}}{\mathrm{R}}\) (d) \(\frac{2 \Delta^{2}}{\mathrm{R}^{2}}\)

Short answer

Answer: The relation between the product of the perpendiculars P1, P2, P3, and the inradius (r), circumradius (R), and side lengths (a, b, c) of the triangle is given by the expression: $$ P_1 P_2 P_3 = \frac{2 \cdot (Area\cdot R)^2}{a \cdot b \cdot c} $$

Step-by-step solution

Express area of the triangle in terms of side lengths and Altitudes (P1, P2, P3)

Recall that the area of a triangle can be expressed in terms of the base, and its corresponding altitude. In this case, we will have three expressions for the area, one using each side as base: $$ Area = \frac{1}{2} \cdot a \cdot P_1 = \frac{1}{2} \cdot b \cdot P_2 = \frac{1}{2} \cdot c \cdot P_3 $$ where \(a,b,c\) are the side lengths of the triangle ABC, opposite to the vertices A,B and C respectively.

Express area of the triangle in terms of semi-perimeter and side lengths (Heron's formula)

Using Heron's formula, we can express the area of the triangle in terms of its semi-perimeter (s) and side lengths: $$ Area = \sqrt{s(s-a)(s-b)(s-c)} $$ where \(s = \frac{a+b+c}{2}\) is the semi-perimeter of the triangle.

Express inradius and circumradius in terms of triangle's area and semi-perimeter

Recall that, the inradius and circumradius of a triangle can be expressed in terms of its area and semi-perimeter: $$ r = \frac{Area}{s} $$ $$ R = \frac{a \cdot b \cdot c}{4 \cdot Area} $$

Calculate the product P1, P2, P3

Using the equations from Step-1, we get the product of the perpendiculars as: $$ P_1 P_2 P_3 = 2\cdot \frac{Area^2}{a \cdot b \cdot c} $$

Use the expressions from previous steps to replace the area in Step 4

Now, we can rewrite the expression for the product of P1, P2, and P3 in terms of the inradius and circumradius by replacing the area, using the expressions from Step 3: $$ P_1 P_2 P_3 = 2\cdot \frac{(r \cdot s)^2}{\frac{a \cdot b \cdot c}{4 \cdot R}} $$ Simplifying the above expression, we get: $$ P_1 P_2 P_3 = \frac{2 \cdot (r \cdot s)^2 \cdot 4R}{a \cdot b \cdot c} $$ We observe that \((r \cdot s)^2 \cdot R = (Area \cdot R)^2\) from the expressions in Step 3. So, the expression becomes: $$ P_1 P_2 P_3 = \frac{2 \cdot (Area\cdot R)^2}{a \cdot b \cdot c} $$

Compare the expression with the given options

Comparing the computed expression with the given options, we see that none of them match the derived expression. Therefore, based on the provided options, there is no correct answer.

Key concepts

Area of Triangle

The area of a triangle is a fundamental concept in geometry. It can be calculated using different formulas depending on the known data about the triangle. One basic approach to find the area is through the formula:\[ Area = \frac{1}{2} \times\text{base} \times\text{height} \]For triangle ABC, by considering each side as a base, the area can be expressed with respect to its corresponding altitude as follows:

• Using side \( a \) as the base: \[ Area = \frac{1}{2} \cdot a \cdot P_1 \]
• Using side \( b \) as the base: \[ Area = \frac{1}{2} \cdot b \cdot P_2 \]
• Using side \( c \) as the base: \[ Area = \frac{1}{2} \cdot c \cdot P_3 \]

In these expressions, \( P_1, P_2, P_3 \) are the altitudes from vertices A, B, and C to the corresponding opposite sides. This demonstrates the relationship between the area's calculation and the altitudes of the triangle.

Heron's Formula

Heron's formula is a useful tool for calculating the area of a triangle when only the side lengths are known. Unlike the base-height method, it doesn't require knowledge of the triangle's height. Heron's formula is expressed as follows:\[ Area = \sqrt{s(s-a)(s-b)(s-c)} \]Here, \( s \) is the semi-perimeter of the triangle, calculated as:\[ s = \frac{a+b+c}{2} \]This formula cleverly uses the semi-perimeter and the differences between the semi-perimeter and each side length. It is particularly useful for triangles where the height isn't readily available, allowing the area to be determined with just the side lengths \( a, b, \) and \( c \).

Inradius and Circumradius

In geometry, a triangle's inradius \( r \) and circumradius \( R \) are important in understanding its other properties. The inradius is the radius of the circle that fits perfectly inside the triangle, while the circumradius is the radius of the circle that passes through all the triangle’s vertices.To find these, we rely on the triangle's area and semi-perimeter:

• The inradius is calculated as: \[ r = \frac{Area}{s} \]
• The circumradius is calculated as: \[ R = \frac{a \cdot b \cdot c}{4 \cdot Area} \]

Understanding these concepts helps in solving problems related to triangle intersection points, such as the orthocenter, centroid, and others, which are crucial in advanced geometric problems.

Altitudes and Sides of a Triangle

Altitudes in a triangle play a key role in various calculations and understanding the triangle's structure. An altitude is a perpendicular line segment from a vertex to the line containing the opposite side. Each triangle has three altitudes corresponding to each side.The relationship between altitudes and the triangle's area is significant as seen in the calculation formulas. The product of altitudes \( P_1, P_2, P_3 \) gives insight into the geometric properties of the triangle.It's fascinating to note that:\[ P_1 \cdot P_2 \cdot P_3 = 2 \cdot \frac{(Area)^2}{a \cdot b \cdot c} \]This remarkable relationship ties together the altitudes, sides, and area, illustrating the interconnectedness of these geometric elements. The simplicity in these equations also shows the elegance of geometry, enabling complex calculations using fundamental triangle properties.