ABCD is a square. X is the mid-point of AB. Y is a point on BC such that BY = \( \frac{1}{3} \) BC. If the area of Δ BXY is equal to 300 cm², find length of diagonal AC.
The problem involves a square ABCD, with specific points X and Y defined on its sides. We are given the area of triangle BXY and need to find the length of the diagonal AC. We will use the properties of a square, the formula for the area of a triangle, and the Pythagorean theorem.
Let the side length of the square ABCD be \(a\) cm.
X is the mid-point of AB. Therefore, the length of BX is half of AB:
\[ \text{BX} = \frac{1}{2} \text{AB} = \frac{a}{2} \]Y is a point on BC such that BY = \( \frac{1}{3} \) BC. Therefore, the length of BY is:
\[ \text{BY} = \frac{1}{3} \text{BC} = \frac{a}{3} \]Triangle BXY is a right-angled triangle with the right angle at B (since ABCD is a square, \(\angle B = 90^\circ\)).
The area of a right-angled triangle is given by \( \frac{1}{2} \times \text{base} \times \text{height} \). For \( \Delta \text{BXY} \), the base can be BX and the height can be BY.
\[ \text{Area}(\Delta \text{BXY}) = \frac{1}{2} \times \text{BX} \times \text{BY} \]Substitute the expressions for BX and BY in terms of \(a\):
\[ \text{Area}(\Delta \text{BXY}) = \frac{1}{2} \times \left(\frac{a}{2}\right) \times \left(\frac{a}{3}\right) \] \[ \text{Area}(\Delta \text{BXY}) = \frac{1}{2} \times \frac{a^2}{6} = \frac{a^2}{12} \]We are given that the area of \( \Delta \text{BXY} \) is 300 cm\(^2\). Set up the equation:
\[ \frac{a^2}{12} = 300 \]Solve for \(a^2\):
\[ a^2 = 300 \times 12 \] \[ a^2 = 3600 \]Solve for \(a\):
\[ a = \sqrt{3600} \] \[ a = 60 \text{ cm} \]So, the side length of the square is 60 cm.
The length of the diagonal of a square with side length \(a\) is given by \(a\sqrt{2}\).
\[ \text{Diagonal AC} = a\sqrt{2} \]Substitute the value of \(a\):
\[ \text{Diagonal AC} = 60\sqrt{2} \text{ cm} \]B) The length of the diagonal AC is \(60\sqrt{2}\) cm.
A) 50\(\sqrt{2}\) cm — This would be the diagonal if the side length of the square were 50 cm. This is incorrect because our calculated side length is 60 cm.
C) 60 cm — This is the side length of the square, not the length of the diagonal. The diagonal is \(a\sqrt{2}\).
D) 50 cm — This is incorrect. It might be chosen if an error was made in calculating \(a\) or if it was confused with the side length of a different square.
The problem asks us to find the length of a room given the total cost of flooring, the cost per square meter, and the ratio of its breadth to length. We will first calculate the area of the room using the total cost and cost per square meter. Then, we will use the given ratio to express breadth in terms of length, substitute these into the area formula, and solve for the length.
Calculate the Area of the Room:
Given:
The area of the room can be calculated by dividing the total cost by the cost per square meter:
\[ \text{Area} = \frac{\text{Total Cost}}{\text{Cost per m}^2} = \frac{3675}{12.50} \]\[ \text{Area} = 294 \text{ m}^2 \]Set up the Ratio of Breadth to Length:
Given that the breadth and length are in the ratio 2 : 3. Let the breadth be \(2x\) and the length be \(3x\).
Formulate the Area Equation:
For a rectangular room, the area is given by the product of its length and breadth:
\[ \text{Area} = l \times b \]Solve for \(x\):
Substitute the expressions for length and breadth, and the calculated area into the area equation:
\[ 294 = (3x) \times (2x) \]\[ 294 = 6x^2 \]\[ x^2 = \frac{294}{6} \]\[ x^2 = 49 \]\[ x = \sqrt{49} \]\[ x = 7 \]Calculate the Length:
Now that we have the value of \(x\), we can find the length:
\[ \text{Length} = 3x = 3 \times 7 \]\[ \text{Length} = 21 \text{ m} \]Correct Option: A) 21 m
This is an analogy question where we need to find the relationship between the first pair of words and apply the same relationship to the third word to find the fourth word. The relationship here is between an animal and its scientific family or classification.
Correct Option: C) Canine
To find the sum, we first need to identify the greatest 3-digit number and the smallest 4-digit number that have 24 as their HCF. A number having 24 as its HCF means the number must be a multiple of 24. We will find the largest multiple of 24 that is a 3-digit number and the smallest multiple of 24 that is a 4-digit number.
Find the greatest 3-digit number that is a multiple of 24:
Find the smallest 4-digit number that is a multiple of 24:
Calculate the sum:
B) 1992 is the sum of the greatest 3-digit number (984) and the smallest 4-digit number (1008) that are multiples of 24, and thus can have 24 as their HCF.
The question asks to identify the highest peak in India. This requires knowledge of the major mountain peaks and their geographical locations and heights.
Correct Option: D) Kanchanjunga
Kanchanjunga (also spelled Kangchenjunga) is the third highest mountain in the world and the highest peak in India. Its elevation is approximately \(8,586 \text{ meters}\) (\(28,169 \text{ feet}\)). It lies on the boundary between Nepal and the Indian state of Sikkim.
The question asks to identify the specific session of the Indian National Congress where the demand for 'Swaraj' (self-rule) was first officially declared. This requires knowledge of key historical events and sessions of the Indian National Congress.
D) 1906
The Calcutta Session of the Indian National Congress in 1906, presided over by Dadabhai Naoroji, is historically significant because it was in this session that the Congress officially declared 'Swaraj' (self-rule) as its goal. This marked a crucial shift in the nationalist movement's objectives.
The question asks to identify the country to which the Ruhr region, a significant industrial area, belongs. This requires knowledge of world geography and major industrial centers.
Correct Option: B) Germany
The question asks about a specific appointment, the Governor of Telangana, made in September 2019. This requires knowledge of current affairs and political appointments from that period.
Correct Option: A) Tamilisai Soundararajan was appointed as the Governor of Telangana on September 1, 2019. She served in this capacity until March 2024.
The question asks to identify the author who used the pen name Madhavikutty. This is a direct factual recall question related to famous Indian authors and their pseudonyms.
Correct Option: C) Kamala Das is the famous author who used the pen name Madhavikutty for her Malayalam writings. She was also known as Kamala Surayya and wrote extensively in both English and Malayalam.
To find the shortest person in the group, we need to establish the relative heights of all individuals based on the given statements. We will use inequality signs to represent "taller than" or "shorter than" and then combine these relationships to form a complete order.
This can be written as: Bella > Shubham
This gives us two relationships:
Combining this with the first statement, we get: Bella > Umesh > Shubham
This means Salman is shorter than Rahul AND Salman is shorter than Shubham.
This can be written as: Shubham > Rahul
From step 2: Bella > Umesh > Shubham
From step 4: Shubham > Rahul
Combining these: Bella > Umesh > Shubham > Rahul
Now, let's incorporate the information from step 3: Salman < Rahul and Salman < Shubham.
Since Shubham > Rahul, and Salman is shorter than Rahul, it automatically means Salman is shorter than Shubham. The critical part is Salman < Rahul.
So, the complete order from tallest to shortest is: Bella > Umesh > Shubham > Rahul > Salman
From the combined order, Salman is at the end, meaning Salman is the shortest.
Correct Option: A) Salman