Triangles

Let the coordinates of points , , and be , , and respectively. Since BC is parallel to the X-axis, the slope of BC is .
In an equilateral triangle, the lengths of all sides are equal. Using geometric properties and calculating the slopes of the sides AB, BC, and CA:

1. The slope of BC is , as BC is parallel to the X-axis.
2. The slopes of AB and CA are the slopes of the other two sides of the triangle.

Since the triangle is equilateral, the slopes of these sides are equal in magnitude but opposite in sign, resulting in slopes of and .

The correct answer is option (A):

A right-angled triangle follows the Pythagorean theorem, which states: where and are the legs and is the hypotenuse. Let's check each option:
1. : , , and , and .
This combination does not form a right-angled triangle.
2. : This satisfies the conditions for forming a right-angled triangle.
3. : This combination satisfies the angle sum property of a triangle and can form a right-angled triangle.
Therefore, the correct answer is (D) None. All combinations are valid for forming a right-angled triangle.

The correct option is (D) : None

The points do not form a triangle when they are collinear. To check for collinearity, we can use the determinant formula. For three points , they are collinear if the following determinant is zero: Substitute the coordinates: Expanding the determinant:

Step 1: Use the formula for the altitude of a triangle.

The area of can be expressed in two ways:

• Using the base and height :
• Using Heron's formula or directly from the sides , , and :

Equating the two expressions for the area:

Simplify:

Step 2: Express in terms of the sides of the triangle.

From the Pythagorean theorem, the hypotenuse of is:

The sine of angle is given by:

Step 3: Substitute into the equation for .

Substitute and into :

Simplify:

Final Answer: The length of is , which corresponds to option .

Step 1: Recall the property of similar triangles.

In two similar triangles, the ratio of any corresponding linear dimensions (such as sides, altitudes, medians, or angle bisectors) is the same as the ratio of their corresponding sides.

Step 2: Apply the property to the altitudes.

Since the ratio of the corresponding sides of the two triangles is , the ratio of their corresponding altitudes will also be .

Final Answer: The ratio of the corresponding altitudes is , which corresponds to option .

We are tasked with finding the length of in the given figure, where , , , and .

Step 1: Use the Basic Proportionality Theorem (BPT).

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Here, , so:

Step 2: Substitute the known values.

From the problem, , , and . Let , so . Substituting into the proportion:

Step 3: Solve for .

Cross-multiply to eliminate the fractions:

Expand the terms:

Rearrange to isolate :

Final Answer: The length of is , which corresponds to option .

Step 1: Recall the relationship between slopes of perpendicular lines.

If the slope of a line is , then the slope of a line perpendicular to it is .

Step 2: Find the slope of the given line.

The general form of a straight line is . The slope of the line is given by:

For the line , and . Thus, the slope of the line is:

Step 3: Find the slope of the perpendicular line.

The slope of a line perpendicular to this one is the negative reciprocal of :

Final Answer: The slope of the line perpendicular to is , which corresponds to option .

Step 1: Recall the formula for the area of a triangle given its vertices.

The area of a triangle with vertices , , and is given by:

Step 2: Substitute the coordinates of the vertices into the formula.

The vertices are , , and . Substituting these into the formula:

Step 3: Simplify each term in the expression.

Expand the terms inside the absolute value:

Substitute these back into the formula:

Step 4: Combine like terms.

Expand the terms:

Combine all terms:

Step 5: Conclude the result.

The area simplifies to:

Final Answer: The area of the triangle is , which corresponds to option .

Correct answer: circumcentre

Explanation:
The circumcentre of a triangle is the point that is equidistant from all the vertices of the triangle.
It is the center of the circumcircle, the circle that passes through all three vertices of the triangle.

Hence, the point equidistant from the vertices is called the circumcentre.

Correct answer: (0, -1)

Explanation:
In a parallelogram, the diagonals bisect each other.
So, the point of intersection of the diagonals is the midpoint of both diagonals. Let the vertices of the parallelogram in order be:
The diagonals are and Find the midpoint of diagonal : Find the midpoint of diagonal : Since both diagonals intersect at the same midpoint, the point of intersection is (0, -1).

Correct answer: 3abc

Explanation:
Let the vertices of the triangle be:

The centroid of a triangle with vertices is given by: Given that the centroid is , we equate: Cube both sides: Use identity: But we will use a simpler identity when :

Hence,

Given: The vertices of the triangle are , , and .

The centroid of a triangle with vertices , , and is given by:

Step 1: Apply centroid formula

Step 2: Solve for

Final Answer: 2

Given: , and the ratio of squared sides is:

Step 1: Understanding similarity property

In similar triangles, corresponding altitudes are proportional to their respective sides.

Step 2: Solve for

Final Answer: 2 : 3

Given: In , , and the given segment lengths:

Step 1: Apply Basic Proportionality Theorem (Thales' Theorem)

Since , by the theorem:

Step 2: Solve for

Final Answer:

Given vertices:

, , and

Step 1: Calculate the side lengths

Using the distance formula:

AB:

BC:

AC:

Step 2: Identify the triangle type

- Since , the triangle is isosceles.

- To check if it is right-angled, we use the Pythagorean theorem:

, which is not true.

So, the triangle is isosceles but not right-angled.

Final Answer: An isosceles triangle

The centroid of a triangle with vertices , , and is given by $ $

In , . We are given that and cm.

By the Basic Proportionality Theorem (also known as Thales' Theorem), we have $ \frac{AD}{AB} = \frac{AE}{AC} \frac{AD}{DB} = \frac{3}{5} \frac{AD}{AB} = \frac{AE}{AC} AE = 2.1$ cm.